Let $X$ be a locally Noetherian scheme defined over an algebraically closed field $k$ and let $Y\subset X$ be a closed subscheme. Suppose there is an algebraic group $G$ acting on $X$ and a subgroup $H\subset G$ acting on $Y$, in such a way that the closed immersion $Y\to X$ descends to a morphism
$$
f:[Y/H]\to [X/G].
$$
Let us assume $f$ is representable and $[Y/H]$ is Deligne-Mumford.
I am looking for a condition (maybe on $X$ and $Y$, but preferably on the group actions) ensuring that $f$ is locally of finite type. For instance I am happy with assuming that the action of $H$ on $Y$ is proper, but I am not sure it helps. Thanks!



Condition for a morphism of stacks to be locally of finite type
Yes. 


Since the question is local in $X$ we can assume that it is a finite type affine scheme. We always have the induced map of stacks $$f:[Y/H]\rightarrow [X/H]\rightarrow [X/G]$$ given by tensoring a locally trivial $H$ torsor with $G$ over $H$. Now we need to show that the fibre product $[Y/H]\times\_{[X/G]} X$ is finite type.


As a stack this is isomorphic to $[(Y\times G)/H]$. The quotient of $Y\times G$ by the action of $H$ always exists in the category of algebraic spaces. This is in Knutson's book, II.6.7. Or see Edidin+Graham's <https://arxiv.org/abs/alg-geom/9609018>, Prop.22: from the proof, the quotient is finite type. By your conditions on $f$, the quotient is then in fact a finite type algebraic scheme.


