Do postdocs in the United States get raises in general? (Is this different for postdocs hired under specific grants vs postdocs hired through departments, such as in mathematics?) If there are raises, how are they determined?
From my experience in the US, it is typical for a postdoc to be on a fixed "short term" contract where there are certainly no "promotion" raises, but might or might not provide cost-of-living increases.
A one-year contract will almost certainly contain no salary increase; for a 2- or 3-year contract, they will be minimal at best.
In my experience (U.S., both East and West coasts) where postdocs are state employees, they will get whatever nominal raise is given across the board to state employees, if such a raise occurs. This is what happened both when I was a postdoc, and it is standard practice for us to build small raises into the budget for postdocs.
In the UK, postdoctoral Research Fellows, just as permanent academic members, are normally entitled to a annual increase of a salary spinal point, up to a maximum of the salary band. The precise arrangements are of course University-specific.