Interestingly, here in India, I have had one of the smoothest transfers from physical to eSIM and between eSIMs to new phones (so far). They have itemized steps to follow, and unlike in 2015-2016, these days, I don't even need to talk to customer support. I have done this while I still had access to my old phone (simpler), and eSIM-ed to the new phone. I have also successfully done it after I had no access to the old phone with Apple's buyback-replacement program when the guy who came to my home gave me the new phone, taking away the old one.
The one key thing that happened is that they sent me confirmations and steps to the email attached to my carrier. Besides that, the security features kick in, where I can make/receive calls, but data/SMS on that number is blocked for the next 24 hours (so, no 2FA and other credentials).