The short answer
Two products, two transfer paths
The word eSIM names the digital SIM technology, not a universal right to move every plan. Your regular carrier line may carry your phone number, calls, texts, and mobile data. A travel eSIM is often a separate, prepaid data product with its own activation and device rules. If both are on the old phone, treat them as two separate moves.
Before you start, keep both phones charged and connected to Wi-Fi. Record the carrier or provider name, account or order reference, the new phone's exact model, and which line you want to move. Avoid erasing the old phone just because apps and photos restored successfully.