Alot of the answer to that would depend on the markers that make up the "signature". Each STRs (at least tracked up to y67) appears to have different rates of mutation. Even that data is over a decade old now, last I checked.
One you examined all that you would have a more informed idea of the possibilities.
One of the problems arises in an STR marker that might mutate one in a 1000 years. Where does the 1000 years start and end? That actually would apply to any of the STRs and their mutation rates.
If the family tree is extensive enough, you might be able to develop an idea of that, if there are differences. You are saying there are not differnces in this case.
The point is, if the signature is made of of STRs that mutate at a snails pace, that would say one thing... If they are made up of STRs that mutate every generation or every other generation, that would say another... If you had a mixture of the tow, you would need to consider forward and then back mutations and the like, etc etc.
In other words, there is not an easy answer. :)