Lorn, it's only going to be active in the plant you put it on. It's not going to move from one plant to another underground. I find dandelions are a royal pain no matter what. It seems like you have to stay vigilant with them to keep them from going to seed. When I spray them with Roundup and because it takes so long to kill them they will end up blooming and going to seed right as they giving their last gasp. I've also used Spectracide and it kills the top growth within a day or so, but it tends to make the tap root angry and they will come back with a vengeance. I really think if at all possible digging them out is the best alternative. Believe me, I know that's always possible. I have a huge peony and lily bed in what was my east pasture and that thing kicks my butt weeding it every spring. This next spring I'm going to use newspaper and heavy mulch to help smother some of the crap out.
Best laid plans.