In contrast to both Anonymous Mathematician and JeffE, I've had a fair amount of conference funding come from a University-wide funding source, and never had anything come directly from my department.
In that case, there wasn't actually a restriction on whether or not the research was done at the university, merely that I was a student there, and that I was presenting research (rather than merely presenting at the conference). As you have a very limited number of times you can apply for travel funding using that mechanism, and it seems the frequency of conference travel increases as you approach graduation (and understandably have more to present), I suspect they're content if a few "slip through".
In terms of @JeffE's statement that a lot of funding comes from advisors, I agree, and then it really is up to the individual advisor. Many I know would be happy to fund a student regardless of where the research originated (especially if the student hasn't had a chance to produce research from their lab yet) for the experience, networking, education, etc.
Publication costs are a somewhat trickier question - honestly, I've never had to publish in something with publication costs, so if I were the new supervisor, I'd argue it was an avoidable expense. If its not, then I'd probably say it's the originating university's problem - their faculty are presumably the ones who would be benefitting from the research hitting the press.
Regardless, I'd at least expect your affiliation to be at your new university.