I am just full of questions this week.
Suppose you transplanted a kidney into someone. That person lives for 10 or 20 more years, then dies. You take out the transplanted kidney, and transplant it into someone else. That person lives 10 or 20 more years, then dies. You take out the transplanted kidney and transplant it into someone else…well, you get the idea.
How long could this continue? Has it ever been tried in animals? Wouldn’t it be interesting to know the typical lifetimes of the various individual organs, and wouldn’t there be strong implications for aging research in general?
Huh, wouldn’t there? 
Part of the problem is that the act of transplantation - removal, transport, sticking it in someone else - can or does cause some damage. Not a lot (hopefully) but enough that I think repeating the process would not be practical.
Also, living does put wear and tear on an organ, which continues to age inside the recipient body. Human cells divide only so many times - with a few exceptions called “cancer”. This is why you put a 20 year old heart in a 60 year old recipient and not vice versa. Eventually, the organ would fail because the cells are old and worn out.
Then, some of the anti-rejection drugs used can cause organ damage long-term. And most folks go through at least a couple episdoes of rejection, where their body attacks the new organ, which also causes some damage.
The upshot is an organ transplanted once is not considered suitable to give to someone else.