Doing research/travel between postdocs (pure mathematics, Europe)
I was unaffiliated for a little over a year between postdocs.  I attended two conferences in that time and didn't list any affiliation.  I also made two research visits, for one of which I received some funding from my host's research grant, and being unaffiliated (or 'an independent researcher', as my host put it) was no obstacle.

This might depend somewhat on country and research area, of course.  And I didn't need a visa for either of the research visits, but I'd be quite surprised if not having an affiliation right now would be an obstacle to getting one (again a little depending on the visa-issuing country).