Matching Theory: Search Time Consider the standard Diamonds coconut economy. One could imagine that people who search longer, have a different likelihood of matching someone to trade with.

Has there been something done akin this (in any part of matching theory)? 

Second Example Say, we want to match liver transplants between donors $d$ and recipients $r$. The longer anyone waits, the higher the likelihood of being matched with someone from the other side.

Hence, we would have a distribution of $r$ and $d$ w.r.t the time that they have already been waiting. And instead of an aggregate matching function $M(r, d)$ we would have something that takes into account these distributions.

What has been done in this area so far?
The paper is a couple of years old by now, but I believe the first people to look seriously at the issues you're thinking of are Akbarpour, et al. 

It might be useful to look at that paper and the handful of others that cite it. The paper was originally called 'Dynamic Matching Market Design', so it may have more citations under that name.