Use olive oil to help get the rust off. Then rise with water and dry very good.
I just rinse them after every ride. Once a week I fill a bucket with hot water and soak them for a few minutes, then dry with a towel or cloth. perfectly clean after!
You could use bit wipes or I just get normal toothpaste and an old toothbrush and scrub the bit.
I use peppermint bit wipes that I got at Dover Saddlery
I just rinse it after each ride, never dirty. But if I forget or whatever and it does get dirty, I just get a dish sponge with a scourer and use some dish liquid and warm water.
As soon as you have finished riding put the bridal into water but only the bit needs to be in the water and the bridal resting on the side and let it soak for 10-20 mins and then with your hands just wash it off
But it in a bucket of hot water