How does the Catholic Church identify and collect relics? I have been told that all major Catholic churches contain within the altar a relic of a saint. Given that saints are not declared until long after the death of the person in question, and many saints die in less than favorable conditions, how does the Church go about verifying and collecting relics?
St. Maximilian Kolbe would be a good example of a saint whom we will never be able to identify any of his first degree relics, since he was burned in a Nazi concentration camp during World War II.  However, there are many 2nd and 3rd degree relics.

Relics are the remains of a saint. They are classified according to degrees. First degree relics are physical remains of a saint. They are most typically bones. Second degree relics are objects or items that touched a saint while he was alive. The relics of Christ's Passion would be second degree relics. Third degree relics include anything that touched other relics. Normally, it is the bones of the saints and the relics of Christ's Passion that are mentioned among the relics.

Catholic Forum - Relics
The perfectly preserved body parts of lots of European saints aren't going to be transferred to the Americas any time soon and we've got a lot of work to do to increase our output of saints.
The Code of Canon Law says:

The ancient tradition of placing relics of martyrs or other saints under a fixed altar is to be preserved, according to the norms given in the liturgical books.

Can.  1237 - §2.
So, what do the liturgical books about the degree of relics for altars?
I skimmed over this liturgical book and you can skim it over yourself, but it doesn't say anything about the degree of the relics.  It does however say that relics of martyrs are preferred and that only recognizable body parts can be placed in sight.  Otherwise, they're put in a box (as a reliquary) under the altar.  The altar must be fixed in place so that everyone knows that the relics are there.
I've got to remember to ask what we've got for a relic in my parish. I hope someone remembers.  Hopefully a priest will show up and give you a better answer.