Let $U \subseteq X$ be an open and $Z := X \setminus U$ its closed complement. I want a sequence $$G_0(Z) \to G_0(X) \to G_0(U) \to 0.$$ However $X, U$ are not quasiseparated and perhaps not even quasicompact. Does this sequence still exist? What should I even mean by $G_0$-theory? Would other types of $K$ Theory have such a sequence?
I'm interested in the case where $X, U, Z$ are Artin Stacks, but I'd be happy to hear words of caution that pertain to schemes as well. What should I be careful of without qcqs? What goes wrong? I assume one takes $G$-theory to be coherent sheaves with quasicompact support, but no evident adaptation presents itself without the "qs" part.