I've cleaned multiple cartridge types at one time when I don't do very many of each brass type. No problems here. Prolly wouldn't do 270 and 280 together...might suck to sort em.
Cleaning beforehand keeps the sizing die cleaner, which is helpful to keep things runnin smooth when working large batches. Since I have a tendency to roll high-power centerfire brass through the sizing die hundreds at a time, I want my brass clean to start. Sometimes I clean twice so the brass looks nice, but I just want it reasonably clean at least for sizing most of the time.