On to SMS messages and this is where I'm reminded one of the reasons I want to escape Apple control. These messages are my data and they are backed up into my iCloud account but can I get at them? Or can I download the iCloud backup and pull them out? Of course not. Cue an hour of hunting around for something which would help me and not try to charge me $20+ for the privilege of extracting my own data.
Finally, there is light at the end of the tunnel. If I manually back up my iPhone via iTunes (like in the Olden Times) then I can extract the SMS messages using AllYourTexts which is simple to use and free. Finally, I have a folder of my message history, freed from the Apple Prison.
I'm expecting the Android import to be significantly easier but even if it isn't I now have the data saved.