Some things have 'full faith and credit' sort of legal language and are good anywhere in the US (or among specific signatories) regardless of which court issued it. Protective orders, for example, are valid anywhere in the US regardless of what state/county/tribal court issued them. Suspended driver's license orders are good among all covenant states (most of them) so if you are suspended in TN you can't get a license in GA either.
Other things do not, and search warrants fall in to that category. Without some agreement in place, they are not valid outside of their originating jurisdiction with some limited exceptions. There are work arounds but what is needed and what works is probably not universal across every state/county. I know for me, I can swear out a search warrant anywhere in the state (all gov't employed cops here have statewide jurisdiction) but would need a judge/magistrate for the correct county to sign off on the warrant.