Some possible ideas:
The War of 1812 was similar to both the American Revolution and the Civil War in that many battles were fought "brother against brother" where close relations were on opposing sides of the same battle, particularly along the New York/Canada borders. Some people seeing the prospect of war chose to move to where other relatives lived so they would not have to take up arms against brothers, cousins, and in-laws; some will have moved simply to be out of the way of the fighting, and others to "put their chips in" with the side they thought would ultimately win the war.
What occupation did Andrew have? Some farming families simply went where good arable land was available cheapest, and of course some merchants and tradesman would see a profit to be had in provisioning one, or both, sides of the conflict.
Of course with the numerous major losses the American suffered in the Niagara Peninsula at the start of the war another option is that the head of family was a soldier wounded and left behind or who had deserted and then sent for their family to join them in Canada. In a similar line of thinking they could have been a sutler or camp-follower who got separated from the supply train of an American invasion force later in the war.