Deep Vein Thrombosis (DVT) Support Group

I was told that once I got to theraputic levels of the medicine (INR between 2 and 3) that the odds of the clots traveling was next to zero but they couldn't tell me how the medicine prevents the clots from moving or breaking off.

Your medical team can tell you best.
Everything I have been told, and found for myself to date, indicates that as clot material gets older it gets more stable. New clot is highest risk. Furthermore, Warfarin, if that is what you have been prescribed, works very slowly. It does not dissolve clots, but simply antagonizes the liver's process of making clotting agents using vitamin K. In that way, the body's own notoriously slow process of reversing clots gets a little help.
One thing is for certain. By being in treatment you are likely not developing newer unstable clots. Therefore at the very least you are much safer than before.
When I first developed a clot in a supposedly superficial and non-life-threatening vein I was sent home from the Dr's untreated. Because of that lack of timely treatment, and I believe only because of that lack of timely treatment, the clots spread downstream and became PE over the course of several weeks. I am glad to see more and more newer cases being treated more aggresively early-on.
Best of luck.