What I often do is fill a little bucket with water and rinse the brush in it. Not pushing it down to the bottom but just swirling it around in the bucket. Then under the faucet I keep it in line with the water (downwards) gently pushing it against my hand which also forms like a cup.
Then go with the hair lines under the faucet with your fingers and ‚massage‘ it a bit so the little reservoir inside the brush will be clean aswell.
Push the water out with your hand gentle and with the hairs. I then take some paper and do the same thing but most of the water will be absorbed then by the papercloth. Then hang it to dry 🙂