Good question.
How about living descendants of Ulysses S. Grant - when was the last time they cracked the tomb open and did a tissue sample?
And how do we know these descendants are really descendants?
I think the answer is "tradition", like you said. It's a little creepy to me, but some of these "remains" were not complete - meaning not all the body parts could be accounted for. Some bones may be where they are said to be - though after all this time, there will be nothing but ashes left - but there was a lucrative market for relics and the more prestigious and wealthy a church was, the better chance they got to get "dibs" on the important relics. Some small insignificant church may only have been able to get Saint So-and-so's baby toe. It's no surprise that Rome laid claim to most of the "biggies". It borders on the macabre, to me, because people come to venerate these remains and relics as if they had supernatural powers. At the resurrection, it's a good thing God knows who is where, that's all I can say. ;o)