Most that I have seen were swamped, but some were straight.
I don't think there is a consistent correlation between period and barrel style (straight vs. swamped) on mountain rifles, other than you wouldn't really expect to see straight barrels in the early period. Most surviving mountain rifles are from the period when commercially made straight barrels were becoming available, yet some mountain gunsmiths apparently continued making swamped barrels right up through the late 19th century.
I think in general there was a tendency toward heavier barrels as time went on, but even that is probably an overgeneralization as well - I have seen some fairly early (early as far as mountain rifles go - pre 1820) guns with barrels that folks today would not consider using for an offhand gun. The weight on many of these guns is surprising when you pick them up - the crisp slender lines tend to hide the scale of the barrel.
Guy