Hey there, PieterD202,
Congratulations on your new iPhone 13 mini!
To make sure all of your mail has transferred to your new device, we suggest first verifying with your email provider that the messages are available on their webmail. If they are, signing into your email account on the iPhone should sync them to the device. You can set an email account up following Add an email account to your iPhone, iPad, or iPod touch - Apple Support.
Delete and recover emails in Mail on iPhone - Apple Support -- When using the term “Archive”, did you archive using the Mail app? If yes, and if you were able to locate the emails missing on your iPhone at your email provider’s site, we recommend deleting and re-adding the email account on your iPhone following the steps provided here: Add and remove email accounts on iPhone - Apple Support
Don't hesitate to respond back if you still need help.
Cheers!