So you’re looking for dropped bars, more versatile tyres and some wider gearing? There’s not much off the shelf but if you’re happy to do some basic bike mechanics, you can definitely roll your own on a reasonable folder with disc or vee brakes and a cassette hub.
Dropped bars are easiest with a folding bike with a T-shaped handlepost - you’ll need a stem replacement such as an Aber Hallo for a 24.5mm diameter bar, or an equivalent by LitePro or similar for a 31mm diameter bar. If the latter, make sure the 31mm diameter centre extends sufficiently from the centre to meet the “stems” on either side.
You’ll need to think about your braking options - if you’re on vee brakes with dropped bars then Tektro RL520s or similar will be needed to handle the longer cable pull, or otherwise Travel Agents are necessary to convert your usual road brake levers or brifters.
In terms of tyres, I’m fairly sure that the average 20” folder can take a 2” tyre e.g. Schwalbe Big Apples and mudguards comfortably. Have a look at BMX tyres if you want to trade grip for puncture proofness.
If you’re in search of wider gearing and plenty of ground clearance without needing a front derailleur, have a look at Microshift and their nine-speed Advent range, which includes a range of shifters and now a super short rear derailleur that can handle an 11-38T cassette.
For me, an ideal build would probably be a Tern Verge frame, 406 wheels with Kenda Small Block 8 tyres , road disc brakes, a wide gravel bar, and the super short Microshift Advent groupset with brifters from the adult range.
For inspiration, this is what I’ve done to a Tern Link D8:
This is with a medium length rear derailleur and a larger 11-46T cassette, but still on Microshift Advent. Clearance isn’t great for hardcore off-road but seems fine on the gravel paths in my corner of the world. I’m currently using a bar end shifter for simplicity, affordability and the fact that I grew up on downtube shifters on road bikes. Tyres are Schwalbe Big Apples.