It is "well-known" that the stack of polarized varieties is an algebraic stack with quasi-compact and separated diagonal.

In particular, if $(X,L)$ and $(Y,M)$ are polarized schemes over a scheme $S$, the Isom-scheme $$Isom((X,L),(Y,M))$$ is separated and quasi-compact over $S$.

I had the feeling that these Isom-schemes should be of finite type and affine over $S$ and surely this is well-known. In other words,

Is the diagonal of the stack of polarized varieties of finite type?

Is it affine?

I did not find an answer in the stacks project. I'm currently looking at work of Rydh and others, but haven't gotten lucky yet.