The main thing top remember about improvisation in traditional music is that traditional music is social. Particularly as regards playing in sessions. So any improvisation has to have regard to what the other players are doing. Yes, improvise around the tune- but it has to fit in with the tune itself, not just the chords, and it has to keep to the feel of the tune.

The result is usually, if you like, micro- improvisations of short duration that return to the main tune fairly quickly- a snatch of harmony, a change of time or a syncopation for a couple of bars. Some players are dab hands at incorporating little quotes from other tunes and making it fit- though others do that by mistake and cause a pile-up.

I find that klezmer lends itself to improvisation much more than Irish (or Scottish or English) music. Swedish music is usually based on harmonies in several parts, so it's fair to slip from one part to another. Certain French sessions I've known hold to the (false) idea that French music is traditionally played one tune at a time, not in sets, so it's almost essential to work in variations to avoid RSI.