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.