'Scheme-theoretic images' of morphisms of stacks

We give criteria for certain morphisms from an algebraic stack to a (not necessarily algebraic) stack to admit an (appropriately defined) scheme-theoretic image. We apply our criteria to show that certain natural moduli stacks of local Galois representations are algebraic (or Ind-algebraic) stacks.
137 pages; final version although not incorporating minor edits made in proof, appeared as Algebraic Geometry 8 (1) (2021) 1--132