For a smooth variety $X$ over a perfect field of characteristics $p$ the sheaf of differential operators is an Azumaya algebra(etale locally is isomorphic to endomorphisms of its center, which is bigger than in char=0), so gives a class $[\mathcal{D}_X]\in Br(X)$ which is always non-zero, because differential operators form a non-trivial Azumaya algebra and is functorial under etale morphisms.

Are there any other non-trivial functorial classes in Brauer group?

Namely, can we choose a class $c_X\in Br(X)$ for every $X$ such that for $f:X\to Y$ we have $c_X=f^*c_Y$ and not all $c_X$ are zero?

Probably, when considering all morphisms, the answer is no. Can something be said for smaller classes of morphisms?

In char=0 I don't even know any examples with reasonably large class of morphisms.

For a smooth variety $X$ over a perfect field of characteristics $p$ the sheaf of differential operators is an Azumaya algebra(etale locally is isomorphic to endomorphisms of its center, which is bigger than in char=0), so gives a class $[\mathcal{D}_X]\in Br(T^*X^{(1)})$ which is always non-zero, because differential operators form a non-trivial Azumaya algebra and is functorial under morphisms of cotangent bundles coming from etale morphisms of the base.

Are there any other non-trivial functorial classes in Brauer group?

Namely, can we choose a class $c_X\in Br(X)$ for every $X$ in some reasonable subcategory $\mathcal{C}\subset \mathrm{Var}_k$ such that for all morphisms $f:X\to Y$ we have $c_X=f^*c_Y$ and not all $c_X$ are zero?

Probably, when considering the whole category, the answer is no. Can something be said for smaller subcategories?

In char=0 I don't even know any examples with reasonably large subcategory.

(Edited thanks to David Ben-Zvi)