You can export just the found records, but you need to first make a Saved Search to filter your records. Then on the record details screen when you select the Export Records or Archive command, Tap Forms will export only those records that match the search rules.
To export only certain fields, it’s a bit more cumbersome on the iOS version. You have to go to the Customize Form screen, then tap on a field in your form, then tap on Field Options, then tap on the switch to turn off the Export Field option.
It’s much quicker to do it on the Mac version since that function is available right on the Export Records window.
Thanks for the detailed instructions. And yes, it is very easy to export (and import) on the Mac. And it is also great that when I have only a mobile device, I can still do an export which, for me, often is a need. So thank you for the functionality on both platforms!