Short answer
To move your eSIM to a new phone, first remove the eSIM from the old phone, then install it again from the app on the new phone.
Before you start
Make sure:
the new phone supports eSIM;
the new phone is unlocked;
you can access the app on the new phone;
you are using the same Apple ID / app account used for the original purchase, if required.
Steps
Remove the eSIM from the old phone.
Install the app on the new phone.
Restore purchases if needed.
Open the app on the new phone.
Go to Settings → Get Link to Install eSIM.
Follow the installation steps.
Important
The eSIM must be removed from the old phone before installing it on a new one.
If you see an error like Activation Link Already Used, Code No Longer Valid, or eSIM Cannot Be Added, it may mean the eSIM is still installed on the old phone or was already used. Contact support with a screenshot of the error.