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I've been trying to find a good reference for a while, it doesn't seem there are many.

In Simplicial Methods for Operads and Algebraic Geometry, Moerdijk and Toën define an étale $n$-algebraic morphisms of (derived, I'll omit this adjective from now on) stacks $f\colon \mathscr{X}\to \mathscr{Y}$ by requiring that for every scheme $U\to \mathscr{Y}$, the fibered product $U\times_\mathscr{Y}^h \mathscr{X}$ admits a smooth $n$-atlas $V\to U\times_\mathscr{Y}^h \mathscr{X}$ (i.e. a smooth $(n-1)$-algebraic epimorphism from a scheme, $n=0$ means representable) such that $V\to U$ is étale.

If we ignore the "derived" everywhere (I don't know if this is bad in any way) and take $n=1$, this would say that a morphism $\mathscr{X}\to \mathscr{Y}$ between algebraic stacks is étale if for every scheme with a map $U\to \mathscr{Y}$ there exists a smooth atlas $V\to U\times_\mathscr{Y}\mathscr{X}$ such that $V\to U$ is étale.

Now for your case, the base change of $\sqrt[r]{\mathscr{L}/\mathscr{Y}}$ along $T\to \mathscr{Y}$ is $\sqrt[r]{L/T}$ where $L$ is the pullback of $\mathscr{L}$, and since $\sqrt[r]{L/T}$ is a $\mu_r$-gerbe over $T$, if $r$ is invertible you have an atlas $U\to\sqrt[r]{L/T}$ such that $U\to T$ is étale.

EDIT: I'm adding a couple of things in response to comments.

If by "your" definition you mean the one with atlases, that has the problem that I outlined in my first comment to your question, that being étale is not smooth-local in source and target. In the definition I wrote down above, you require something for all maps of schemes to the target, and the existence of an atlas of the pullback that does what you want.

In the DM case this is easier, because being étale is étale local on source and target, and you can just ask it for one pair atlases and it will be true for any.

About affine morphisms, the only definition I've ever seen implies representability. You might be interested in cohomologically affine morphisms (see http://arxiv.org/pdf/0804.2242v3.pdf )

About finite, you can either say "representable finite" (and I think this is the standard) or quasi-finite (that is a condition on geometric points) and proper. Proper is separated (the diagonal, which is representable, is proper), finite type and universally closed (that you define using the associated topological spaces). Alternatively, there is a valuative criterion for properness.

Maybe you've already seen those, but let me also point out this discussion http://math.stackexchange.com/questions/1104790/this-property-is-local-on-properties-of-morphisms-of-s-schemes and this http://stacks.math.columbia.edu/tag/04QW and the neighboring sections of the stacks project.

I've been trying to find a good reference for a while, it doesn't seem there are many.

In Simplicial Methods for Operads and Algebraic Geometry, Moerdijk and Toën define an étale $n$-algebraic morphisms of (derived, I'll omit this adjective from now on) stacks $f\colon \mathscr{X}\to \mathscr{Y}$ by requiring that for every scheme $U\to \mathscr{Y}$, the fibered product $U\times_\mathscr{Y}^h \mathscr{X}$ admits a smooth $n$-atlas $V\to U\times_\mathscr{Y}^h \mathscr{X}$ (i.e. a smooth $(n-1)$-algebraic epimorphism from a scheme, $n=0$ means representable) such that $V\to U$ is étale.

If we ignore the "derived" everywhere (I don't know if this is bad in any way) and take $n=1$, this would say that a morphism $\mathscr{X}\to \mathscr{Y}$ between algebraic stacks is étale if for every scheme with a map $U\to \mathscr{Y}$ there exists a smooth atlas $V\to U\times_\mathscr{Y}\mathscr{X}$ such that $V\to U$ is étale.

Now for your case, the base change of $\sqrt[r]{\mathscr{L}/\mathscr{Y}}$ along $T\to \mathscr{Y}$ is $\sqrt[r]{L/T}$ where $L$ is the pullback of $\mathscr{L}$, and since $\sqrt[r]{L/T}$ is a $\mu_r$-gerbe over $T$, if $r$ is invertible you have an atlas $U\to\sqrt[r]{L/T}$ such that $U\to T$ is étale.

EDIT: I'm adding a couple of things in response to comments.

If by "your" definition you mean the one with atlases, that has the problem that I outlined in my first comment to your question, that being étale is not smooth-local in source and target. In the definition I wrote down above, you require something for all maps of schemes to the target, and the existence of an atlas of the pullback that does what you want.

In the DM case this is easier, because being étale is étale local on source and target, and you can just ask it for one pair atlases and it will be true for any.

About affine morphisms, the only definition I've ever seen implies representability. You might be interested in cohomologically affine morphisms (see http://arxiv.org/pdf/0804.2242v3.pdf )

About finite, you can either say "representable finite" (and I think this is the standard) or quasi-finite (that is a condition on geometric points) and proper. Proper is separated (the diagonal, which is representable, is proper), finite type and universally closed (that you define using the associated topological spaces). Alternatively, there is a valuative criterion for properness.

Maybe you've already seen those, but let me also point out this discussion https://math.stackexchange.com/questions/1104790/this-property-is-local-on-properties-of-morphisms-of-s-schemes and this http://stacks.math.columbia.edu/tag/04QW and the neighboring sections of the stacks project.

I've been trying to find a good reference for a while, it doesn't seem there are many.

In Simplicial Methods for Operads and Algebraic Geometry, Moerdijk and Toën define an étale $n$-algebraic morphisms of (derived, I'll omit this adjective from now on) stacks $f\colon \mathscr{X}\to \mathscr{Y}$ by requiring that for every scheme $U\to \mathscr{Y}$, the fibered product $U\times_\mathscr{Y}^h \mathscr{X}$ admits a smooth $n$-atlas $V\to U\times_\mathscr{Y}^h \mathscr{X}$ (i.e. a smooth $(n-1)$-algebraic epimorphism from a scheme, $n=0$ means representable) such that $V\to U$ is étale.

If we ignore the "derived" everywhere (I don't know if this is bad in any way) and take $n=1$, this would say that a morphism $\mathscr{X}\to \mathscr{Y}$ between algebraic stacks is étale if for every scheme with a map $U\to \mathscr{Y}$ there exists a smooth atlas $V\to U\times_\mathscr{Y}\mathscr{X}$ such that $V\to U$ is étale.

Now for your case, the base change of $\sqrt[r]{\mathscr{L}/\mathscr{Y}}$ along $T\to \mathscr{Y}$ is $\sqrt[r]{L/T}$ where $L$ is the pullback of $\mathscr{L}$, and since $\sqrt[r]{L/T}$ is a $\mu_r$-gerbe over $T$, if $r$ is invertible you have an atlas $U\to\sqrt[r]{L/T}$ such that $U\to T$ is étale.

EDIT: I'm adding a couple of things in response to comments.

If by "your" definition you mean the one with atlases, that has the problem that I outlined in my first comment to your question, that being étale is not smooth-local in source and target. In the definition I wrote down above, you require something for all maps of schemes to the target, and the existence of an atlas of the pullback that does what you want.

In the DM case this is easier, because being étale is étale local on source and target, and you can just ask it for one pair atlases and it will be true for any.

About affine morphisms, the only definition I've ever seen implies representability. You might be interested in cohomologically affine morphisms (see http://arxiv.org/pdf/0804.2242v3.pdf )

About finite, you can either say "representable finite" (and I think this is the standard) or quasi-finite (that is a condition on geometric points) and proper. Proper is separated (the diagonal, which is representable, is proper), finite type and universally injective (that you define using the associated topological spaces). Alternatively, there is a valuative criterion for properness.

I've been trying to find a good reference for a while, it doesn't seem there are many.

In Simplicial Methods for Operads and Algebraic Geometry, Moerdijk and Toën define an étale $n$-algebraic morphisms of (derived, I'll omit this adjective from now on) stacks $f\colon \mathscr{X}\to \mathscr{Y}$ by requiring that for every scheme $U\to \mathscr{Y}$, the fibered product $U\times_\mathscr{Y}^h \mathscr{X}$ admits a smooth $n$-atlas $V\to U\times_\mathscr{Y}^h \mathscr{X}$ (i.e. a smooth $(n-1)$-algebraic epimorphism from a scheme, $n=0$ means representable) such that $V\to U$ is étale.

If we ignore the "derived" everywhere (I don't know if this is bad in any way) and take $n=1$, this would say that a morphism $\mathscr{X}\to \mathscr{Y}$ between algebraic stacks is étale if for every scheme with a map $U\to \mathscr{Y}$ there exists a smooth atlas $V\to U\times_\mathscr{Y}\mathscr{X}$ such that $V\to U$ is étale.

Now for your case, the base change of $\sqrt[r]{\mathscr{L}/\mathscr{Y}}$ along $T\to \mathscr{Y}$ is $\sqrt[r]{L/T}$ where $L$ is the pullback of $\mathscr{L}$, and since $\sqrt[r]{L/T}$ is a $\mu_r$-gerbe over $T$, if $r$ is invertible you have an atlas $U\to\sqrt[r]{L/T}$ such that $U\to T$ is étale.

EDIT: I'm adding a couple of things in response to comments.

If by "your" definition you mean the one with atlases, that has the problem that I outlined in my first comment to your question, that being étale is not smooth-local in source and target. In the definition I wrote down above, you require something for all maps of schemes to the target, and the existence of an atlas of the pullback that does what you want.

In the DM case this is easier, because being étale is étale local on source and target, and you can just ask it for one pair atlases and it will be true for any.

About affine morphisms, the only definition I've ever seen implies representability. You might be interested in cohomologically affine morphisms (see http://arxiv.org/pdf/0804.2242v3.pdf )

About finite, you can either say "representable finite" (and I think this is the standard) or quasi-finite (that is a condition on geometric points) and proper. Proper is separated (the diagonal, which is representable, is proper), finite type and universally closed (that you define using the associated topological spaces). Alternatively, there is a valuative criterion for properness.

Maybe you've already seen those, but let me also point out this discussion http://math.stackexchange.com/questions/1104790/this-property-is-local-on-properties-of-morphisms-of-s-schemes and this http://stacks.math.columbia.edu/tag/04QW and the neighboring sections of the stacks project.

I've been trying to find a good reference for a while, it doesn't seem there are many.

In Simplicial Methods for Operads and Algebraic Geometry, Moerdijk and Toën define an étale $n$-algebraic morphisms of (derived, I'll omit this adjective from now on) stacks $f\colon \mathscr{X}\to \mathscr{Y}$ by requiring that for every scheme $U\to \mathscr{Y}$, the fibered product $U\times_\mathscr{Y}^h \mathscr{Y}$ admits a smooth $n$-atlas $V\to U\times_\mathscr{Y}^h \mathscr{Y}$ (i.e. a smooth $(n-1)$-algebraic epimorphism from a scheme, $n=0$ means representable) such that $V\to U$ is étale.

If we ignore the "derived" everywhere (I don't know if this is bad in any way) and take $n=1$, this would say that a morphism $\mathscr{X}\to \mathscr{Y}$ between algebraic stacks is étale if for every scheme with a map $U\to \mathscr{Y}$ there exists a smooth atlas $V\to U\times_\mathscr{Y}\mathscr{X}$ such that $V\to U$ is étale.

Now for your case, the base change of $\sqrt[r]{\mathscr{L}/\mathscr{Y}}$ along $T\to \mathscr{Y}$ is $\sqrt[r]{L/T}$ where $L$ is the pullback of $\mathscr{L}$, and since $\sqrt[r]{L/T}$ is a $\mu_r$-gerbe over $T$, if $r$ is invertible you have an atlas $U\to\sqrt[r]{L/T}$ such that $U\to T$ is étale.

I've been trying to find a good reference for a while, it doesn't seem there are many.

In Simplicial Methods for Operads and Algebraic Geometry, Moerdijk and Toën define an étale $n$-algebraic morphisms of (derived, I'll omit this adjective from now on) stacks $f\colon \mathscr{X}\to \mathscr{Y}$ by requiring that for every scheme $U\to \mathscr{Y}$, the fibered product $U\times_\mathscr{Y}^h \mathscr{X}$ admits a smooth $n$-atlas $V\to U\times_\mathscr{Y}^h \mathscr{X}$ (i.e. a smooth $(n-1)$-algebraic epimorphism from a scheme, $n=0$ means representable) such that $V\to U$ is étale.

If we ignore the "derived" everywhere (I don't know if this is bad in any way) and take $n=1$, this would say that a morphism $\mathscr{X}\to \mathscr{Y}$ between algebraic stacks is étale if for every scheme with a map $U\to \mathscr{Y}$ there exists a smooth atlas $V\to U\times_\mathscr{Y}\mathscr{X}$ such that $V\to U$ is étale.

Now for your case, the base change of $\sqrt[r]{\mathscr{L}/\mathscr{Y}}$ along $T\to \mathscr{Y}$ is $\sqrt[r]{L/T}$ where $L$ is the pullback of $\mathscr{L}$, and since $\sqrt[r]{L/T}$ is a $\mu_r$-gerbe over $T$, if $r$ is invertible you have an atlas $U\to\sqrt[r]{L/T}$ such that $U\to T$ is étale.

EDIT: I'm adding a couple of things in response to comments.

If by "your" definition you mean the one with atlases, that has the problem that I outlined in my first comment to your question, that being étale is not smooth-local in source and target. In the definition I wrote down above, you require something for all maps of schemes to the target, and the existence of an atlas of the pullback that does what you want.

In the DM case this is easier, because being étale is étale local on source and target, and you can just ask it for one pair atlases and it will be true for any.

About affine morphisms, the only definition I've ever seen implies representability. You might be interested in cohomologically affine morphisms (see http://arxiv.org/pdf/0804.2242v3.pdf )

About finite, you can either say "representable finite" (and I think this is the standard) or quasi-finite (that is a condition on geometric points) and proper. Proper is separated (the diagonal, which is representable, is proper), finite type and universally injective (that you define using the associated topological spaces). Alternatively, there is a valuative criterion for properness.