DOT's website has an address to which complaints need to be directed.

They will generally send an acknowledgment (and some stupid government pamphlet on consumer rights...) within a 3-4 week period.

Normally DOT passes along the complaint to various regulatory affairs folk within the airline. Sometimes that will result in compensation and/or an equitable resolution, sometimes not. IIRC, the regulatory affairs folks need to respond back to DOT and indicate how they resolved it.

I've probably filed 2-3 complaints with DOT over my 25 years of heavy traveling. In two cases I've received compensation (in one case it was $200 vouchers each for me and my companion), in the other it was some kind of voucher and a promise to treat me better. In the third case, I never heard anything back other than the acknowledgment. In both cases where I got vouchers, I was top-tier in the FF program.

YMMV.