Transfer app data (settings, etc) between phones with different Android version

As title says, can anyone recommend a way to transfer app data and settings between two phones with different Android versions? I'm switching from my old Z1 compact with LP 4.4.4. installed to a Z5 compact with MM 6.0.1 (32.2.A.0.253), and I'd like to make the switch as smooth as possible (e..g not having to log in into all various apps all over again). Is this possible, or will this involve some manual work due to the different Android version?

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