I grew up in south west Scotland, and yes, we used flitting when someone moved house. The phrase moonlight flit was also used, and meant, as others have said, someone running away and hiding - usually from debts, but I suppose it could also have meant in the way grannyactivist explained it. I'd not really thought about it that way to be honest.
When mum and I were ready to move from the first home we had in England to the next one, and I told friend we were flitting, I got the oddest of looks - not for the first time though. Many words and phrases I used met with bemusement from my English friends 