Now that iTunes doesn't manage the apps on my iPhone or iPad, how to I transfer my FM database to these iOS devices?
You can email it to yourself then open on device or you can use iCloud Drive, Dropbox, or cloud type service.
In FMGo, click the device icon, then click the download icon (folder with down arrow), then select the location (iCloud Drive), then select your file.
You can use Airdrop: https://support.apple.com/en-us/HT203106
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