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There is an apple description of how to do this(but I could not readily find it).

So... I’ll tell you what I did. I went from a late 2013 MBP to a 2021 M1Pro. The 2013 had a 256 GB SSD and the new one has a 512 gb SSD. The late 2013 is using OS Mojave.

First I purchased a new one TB HDD WD USB drive from Walmart. I formatted it using Guid APFS without encryption(don’t want to make it any more complicated than it has to be.) You could use another older one and re-format it.

Next I made one Time Machine backup using the 2013, .

Then I attached the WD to the new M1Pro. I needed a USBa to USBc dongle.

I opened the screen, and very shortly was asked(because the TM backup was attached to a USBc port) whether I wanted to migrate using the existing TimeMachine backup. I clicked yes. In about 50 minutes, my new M1Pro looked almost identical to my previous machine with all the same internet connections, and passwords.

I did not need to have the same OS in both machines using this method.

Hope this helps.

Are you saying that you upgraded to a mac with a smaller SSD than the one that was on the original mac?

If this is the case, and you want to keep all the files, I would suggest you buy an inexpensive 2TB HDD(not SSD) and off load all the files you want to keep, and then erase your current time machine back up, reformat it to APFS, and make sure that the whole combination is less than the total GB’s in your current drive. Make a brand new Time Machine backup and then use this time machine back up to migrate.