Will a 3x10 or 3x9 front derailleur work on a 6/7 speed chain?
Generally yes, the front mech can work acceptably with other sizes of chain.
Since you will have a thicker chain than expected, the front derailleur cage may need to be widened a little to reduce chain rub.
The actuation ratio is between your shifter and the chainrings, so a friction shifter will work perfectly, and an indexed shifter should be acceptable.  If you're mixing shifter/chainrings then things could get odd but that's more likely above 10speed.  Below 10 speed, the chainring spacing generally was the same.
Remember the front shifter is more of a &quot;brute force&quot; mechanism having to move the chain under tension, compared to the rear shifter whose chain is mostly slack.  So the front shifter has to deal with higher pressures and tensions.

I have an old MTB that was 15 speed, (3x5) which now has a 9 speed rear cassette/derailleur/chain.  The front mech is still the original triple, on the original chainrings.  All I had to do was close the cage a bit to improve shifting.  I do use friction shifter on the left hand which makes trim very flexible.