After refreshing my dim memory of the War of 1812, it looks as though Gen. Jackson took advantage of the Battle of New Orleans being close to Florida and decided to beat up the Spaniards until they decided they’d sell the Sunshine State, so it was a double victory for the US.
From Canada’s viewpoint, it shored up Canadian identity and placed things on a more stable basis.
A few more Yanks drifted north later. I had an ancestor who fought at both the tail end of Lexington and then Bunker Hill, and quite a few other places who had a daughter who settled in Upper Canada in 1816 with her husband, and he ended up joining them about a decade later after being rejected for a military pension multiple times. I doubt they had a thorough immigration interview at the time.
I think all the daughter’s children ended up back in the States. I’m descended from one of his sons who skipped Ontario and headed to Michigan instead.
Anything to get clear of Massachusetts