I've wondered if a bug zapper would work.

Thing is if they are not sealed inside the bin (not a tight seal), even if you kill them all flies from outside will get right back in and repopulate. I'm 18 floors up and when my bin's on the balcony it attracts flies (and yes that scares me :P )

Maybe do what you can to keep the numbers down and have some traps inside.

I thought houseflys were not supposed to go above 10 feet thus the hanging of meat high to age. The old wives tale must not be correct. Or maybe that is mosquitoes.

For Fungus gnats I suggest DDT. Fungus gnats are evil.

Hopefully what works for fruit flys will also work for fungus gnats.

We are going to try a multi pronged approach. We are going to hit them with everything we got all at once for the win.

While covering with bedding is a great idea... I have seen the little buggers hatch and pass on up out of that covering of bedding.

You may want to consider temporarily freezing and blending in no particular order their food.

Food can also be wrapped in newspaper or put into a bag with a rolled closed top. This is added to the bin.

A shear curtain type material can be put on top of the bin.

At night the computer screen attracts them. Or any window with light.

I am now used to a quick scan across the top of my wine glass before each sip. A quick dip of a finger tip targeted into the glass will find a drop of wine with the victim captured.

Maybe stop adding any food at all for a few weeks. Freeze it for later.

Fruit flys seem attracted to the top of pillars. Especially white pillars. A glass you never expect to drink out of with an inch of your finest wine, believe me they always go for the good stuff first, and a well mixed in drop of detergent place on top of the pillar will over the course of a month attract and drown an inch of fruit flys. Unfortunately they do not swarm in immediately.

It is fun to watch and very satisfying to see each fruit fly that goes for the big swim. Don't tell PETA.

Now that would be a science experiment. Do fruit flys go for the $50 bottle instead of the $5 of wine. :-) A new wine rating system.

On the other hand even the fruit flys are helping to do the work of composting and getting the material good and yummy for the worms. As long as their little bodies die inside the compost bucket the value stays in.

Apple-cider vinegar traps for fruit flies and yellow-colored sticky fly paper (sheets or rolls), for fungus gnats.

I'm in N. Cal and keep my worms in unglazed terra cotta planters. Helps to regulate heat and easy to make a fly-proof cover held on tight with bungee cords (like a drum). One of my planters got an infestation once, I just left the cover on a few weeks (stopped feeding and only watered with sprinkling can) and when I checked they were gone. Make sure you get a planter with a lip on it that will hold the bungee cords in place (you'd be surprised how many have that feature, classical look, how did the Romans know about bungee cords?).

I had that problem with fungus gnats especially bad for my bins inside the house. There were many gnats so I tried Gonat spray (cedar), more newspaper and finally food grade dioteneous earth. After 1 or 2 dustings of diatoneous earth over a month problem solved and no return of the gnats in 6 months. Apparently the "earth" acts like tiny razorblades and cuts them up though doesnt harm worms.