@griffin: That's talking about the switch in roots, whereas I think that this is asking about the switch in intermediates. They're kind of related, but as both R3 and X3 are signed by both roots it's really a different question.
My understanding is that at some point leaf certificates will be signed by R3 instead of X3, and that it'll just switch over at some point "soon", but I don't remember seeing a specific date for it yet.
As long as your client is downloading the full chain from the ACME server, you shouldn't need to do anything special. (And at any point Let's Encrypt might need to switch to their backup X4 or R4 without further notice, or issue a new intermediate, and you shouldn't need to do anything specific for any of those cases.)
The alternate link is just for a different root, not for a different intermediate. One day, some or all certificates will just start getting signed by R3 and that's it. The alternate link for different roots will work the same regardless of which intermediate is being used.