They go by the serial number on the sight. YMMV but this is my experience.
1) I bought a TA44-R4 at a gun show. I got a good deal on it because the tritium was out. I contacted JoLeene at Trijicons cust. service dept. and told her my tale.( Used, no warranty card, no tritium etc.) She researched the serial number and said it was in warranty, send it in and they would inspect it. I paid for shipping back to Trijicon and they put in new glass, new reticle and a new tritium lamp in it and sent it back. The warranty then started over. I believe that they would have sent a UPS pick up had I been the original owner and had a card on file. Thats about the only difference.
It took about three weeks to get it back. Its still running strong and sits on a COLT 6530 that my wife shoots.
2) I bought a used TA-50-2 with a chipped rear glass on it. Once again I called Joleene and she researched the serial number. It was in warranty also so she said send it in. I actually sold it the way it was but the purchaser sent it in to be repaired. I dont know how long it took.
This was before the war started. I dont know how fast their warranty service is since they have these big Govt orders.