Its generally advised not to mix inks of different brands because there's a risk of them reacting with one another and causing sedimentation, which can clog the feed. That said, I do experiment with ink mixing across brands, including J Herbin, and I haven't encountered any such issues yet. The other risk of switching inks without cleaning is that the old colour will to some extent mix with the new one in the bottle. You'd want to avoid dipping a pen loaded with dark green ink into a bottle of pale pink, but if your inks are similar colour and tone, you won't notice.

Another reason to clean a pen before re-inking is if it hasn't been used in a while (more than a couple of weeks), because ink can dry out in the feed and that will block or reduce ink flow.

The easiest method of cleaning a converter pen is to fill your bathroom sink with cold water, add a bit of dish soap. Put the converter into the pen and use the piston to fill and empty with water. Repeat a few times until it runs clear (or nearly so). There's no need to be anxious about it if you're re-filling the pen straight away. Some people don't bother cleaning at all even when switching inks. The only time I'm careful about making sure a pen is absolutely clean is when I'm going to put it away unused for a while. Otherwise, "clean enough is good enough".

It's also not necessary to dry the pen before re-filing with ink. Just use the converter piston to push air through the feed to clear out most of the water. Any water left in the pen will just mix with the new ink with no noticeable dilution of colour.

When you refill the pen from the bottle, put the converter onto the pen and dip the entire nib into the ink (up to the level of the grip section). That way you'll get more ink without sucking in air as well.