That last part isn't strictly true; and only two of the four jabs use DNA anyway.
What happens (British Medical Journal article on mice) is that the injection has some small proportion of the lipid-enclosed mRNA carried throughout your body by the lymph and/or blood, to include passing the blood-brain barrier.
The mRNA enters cells which then put a copy of the spike protein sticking out of their surfaces.
If some of the cells are in the lining of the bloodstream, this may cause the clotting.
Also, I spoke to a high-end vascular MD about this yesterday. His suggestion was that the spike protein interferes with anti-clotting proteins: an interesting model I haven't seen ANYONE suggest yet.
The spike protein is what causes all the damage. Clotting is the body’s defense trying to plug all the leaks. They inject millions of these things into your body. Look ate the VAERS reports. Clotting, bleeding, meningitis, heart problems, etc, etc all on a massive scale. These are not rare nor isolated.