Do universities sponsor a student's conference travel and publication costs if the research on the paper has been done elsewhere? For example, a student may have published a paper during a master's at university 1 and may currently be pursuing PhD at university 2. Similarly will the second university bear the cost of the journal publication? Are there any caveats involved (eg., indication of university's name in the paper, related department, etc)?
On the other hand, will the first university sponsor one of its alumni for the travel? (This looks unlikely to me.)
At least in the cases I'm familiar with (math departments at US research universities), funding for travel is more likely to come from an individual department than from a university-wide source, so this would be a matter of department policy. That's a little less true for publication costs, because some universities have started funds to pay for gold open access, but everyone is setting it up a little differently so I don't think one can announce a general rule.
At the department level, this really varies. One department I used to be at provided a certain amount of travel funding to each student, to be used however they wanted (as long as it was for academic purposes, of course). I think that's a little unusual, and many departments don't make a decision until the student applies for funds.
If you are in a field in which these costs are an important part of publication (not pure mathematics, for example), then this is probably the wrong question. In such fields, the funding usually comes from the advisor's grants, rather than the university.