It depends on the nature of the complaint as well as how the other party decides to respond. Here are my two experiences with this and they are not a representation of all that FINRA does in such situations.
1) In 1987, I had a $15 covered call position on Bear Stearns that expired two points in-the-money. I should have been assigned. The next trading day was the crash of 1987 and it took Paine Webber 8 days to figure out whether I had been assigned or not. By then, BSC had dropped to about $13. I had been unable to defend the position if indeed I still owned it. I indicated that I was going take take them to arbitration and they chose to settle, crediting my account with $200 per 100 shares.
2) During the Internet craze, I was dealing with a number of firms, one of which was a "Maggot Mile" broker (located in Boca Raton Florida) who had a string of IPOs that opened for trading on their first day at multiples of the IPO price. I knew what I was dealing with. I did 3 of them and for the last one, the broker refused to execute a trade to sell the balance of a profitable equity position (brokers lose allocation in future deals if their clients are flippers). My loss was the gains since I had sold some shares already. Shortly thereafter, the SEC closed the firm down due to illegal practices and poof, profits gone,
I filed a complaint against the broker and the firm. I had no expectation of receiving remuneration since the firm was kaput. I just wanted to put a "Yes" violation on the broker's license (I don't know the specific terms). It was all handled by mail. I provided a written statement and other documentation and then each party responded back and forth. After about 8 months, I received a verdict in my favor, 50% against the broker and 50% against the brokerage firm. In order to get a job at a new firm, the broker had to settle with me. Found money.
FINRA has a process for arbitration:
Don't bother with a complaint unless you can provide clear cut evidence of malfeasance.