I think they came every way possible.
My distant cousins, the Paleo-Eskimo, got covered by the more recent coat of paint, and that was long after the "land bridge" was so long gone it had vanished from folklore, so, I'd say, they primarly got here by water, and there were a lot of "theys" involved, a sort of sine wave of intensity, of different arrivals.
I'll even go so far as to say that we don't know nearly enough about the depth of prehistory and primatology in the Americas, important to know when one is a multiregionalist.