How to change ownership of a proxy contract

I am in the process of changing the setup of the smart contracts for my project to a proxy - implementation setup. I have used this tutorial and it works well:

However, I then encountered an issue with the ownership of these contracts. My original contracts were partly using ownable for example to restrict minting of new tokens to the owner.

I have tried to google the precise problem, but it seems that I have not yet fully understood the concept and did not manage to do it yet. To be precise, I am not sure if the right to upgrade the implementation contract needs to be given to the address as the "owner" role, which I use to control other functions.

Many thanks in advance for your help,