I typically make multiple small batches in one evening. Washing and drying each tool, mixing container, etc., between batches would triple my time investment, so instead, I very carefully wipe down everything with paper towels until I can no longer see any remnants or streaks. Then, I start a new batch.
The only exception is the dishes I measure out my fragrance oils to. Those are washed and dried because FOs can really cling and affect the next FO.
Everything is washed, dried, and put away when my entire soaping session is done.
I have found that even a trace amount of soap batter will start to accelerate the next batch, and the designs I'd wanted to achieve may not be possible. I've also noticed that if I let my oils and lye mixtures get below a certain temperature, I may start to get graininess while stick-blending the batter. So, I plan the order of my pours based on the behavior of the FO, whether it accelerates or decelerates, in an effort to compensate.
It's a tradeoff for sure, and not always a predictable experience, especially with new-to-me FOs, but worth it to me for the time-saving.