Will Apple care transfer from iPhone 11 to a new iPhone 13

No, but you can cancel the AppleCare plan on your iPhone 11 and buy a new plan for your new phone. See: Cancel or change the payment method for your AppleCare plan - Apple Support and see the following from AppleCare Products - iPhone - Apple

How to buy

1. Purchase either AppleCare+ product with your new iPhone.
2. Or buy it within 60 days of your iPhone purchase:

• • On your device (go to Settings > General > About and choose AppleCare+ Coverage Available)
  • Online (requires you to verify your serial number and run a remote diagnostic)
  • At an Apple Store (inspection of iPhone and proof of purchase are required)
  • By calling 800-275-2273 (requires you to run a remote diagnostic and provide proof of purchase)

No, you will have to purchase coverage for the 13 Pro. You can cancel coverage on the 11 for a partial refund→Cancel or change the payment method for your AppleCare plan - Apple Support