They sure shouldn't "swing around" at all!!
Most are (and should) be implanted partially beneath your pectoral muscle and never "move around." Also depending on how much breast tissue one has already, you might not even feel them from the outside.
I can feel mine from the outside on the bottom, but that is because I have little breast tissue and thin skin. It is also my body and so I am more sensitve about what I feel in it. I can kinda feel them when I sleep on my tummy, it is like sleeping on two big soft balls in your boobs. Weird, but not unpleasant. And they have never felt heavy. I was preparing myself to have a heavy sensation on my chest after surgery (4mos ago) but, that was never the case.
Like any adjustments in life, you get used to it.