The Old Horse is a bit of a case apart. The "mast horse" tradition ran in a band down from North Yorkshire through Derbyshire and Cheshire into North Wales, and was little known outside those areas, though there were other kinds of "horse" in the South of England and in Ireland. The song, however, was widely printed on broadsides and has been found (though without the attendant custom) all over the country. Additionally, parts of it seem to have been adapted for the sailors' song of the same name, though that relates to a completely different custom; examples found in the USA seem to be from the maritime branch.

As you suggest, these kinds of customs tended to be local and didn't usually export well. Families might emigrate together, communities less commonly. The Mummers' Play, though, seems to have spread largely via chapbooks to begin with, and was to a degree standardised, so may have been more widely recognisable -and hence acceptable- among migrants from different areas; and it is Mumming and the Luck-Visit tradition that turn up most often in exported form. Greg has mentioned the Caribbean; there is also quite a lot in Newfoundland. Mumming in the USA is rather a different thing, and where it resembles the British form it appears, like Morris in the USA, to be a recent revival rather than a transmitted tradition.

That's getting well outside my area, though, so you'll be better off looking at the website of the  Traditional Drama Research Group. See also their "Americas" links page:  Folk Play Links - Regional & Performers' Websites