GIT-like stability criteria on algebraic stacks

To find stability conditions for moduli problems one usually proceeds in two steps:
first one makes an educated guess and second one proves that this guess coincides
with a stability criterion coming from GIT. We would like to give a criterion that can
help to avoid the guessing procedure and explain why this can often be used to find
substacks of moduli problems that admit separated coarse moduli spaces. This was
motivated by questions on the cohomology of moduli spaces of bundles with various
kinds of extra structure.