It's a good question and I think it depends on both the client and the server. Windows (IIS etc) for instance will doggedly serve the R3 > DST Root CA X3 chain (regardless of the client used or chain that was installed) unless you disable that version of the R3 but the general expectation is that once the R3 actually expires that will correct itself (in my 'future' testing I needed a server reboot).
The client side is different and depends on how the client reacts when it sees the expired R3, and whether it can build an alternative chain using it's local trust store or other OS features.
There is also this thread but it's too big to get concrete information from now: Help thread for DST Root CA X3 expiration (September 2021)
I believe the conventional wisdom is that it will all just work, but I expect at least a few hiccups along the way.