This thread has gotten very weird. It's probably best practice to pop a bead, wipe out the grody mess of sealant and whatever nastiness it is that turns it brown, and refill directly into the tire. Then pop the bead back on. If you're in a shop and have a way of holding the wheel by the axle this takes less time than injecting it through the valve in my experience. I really want to get a Park WH-1, but it's kinda expensive. You can do it with a bike in a stand, but that usually leaves the tire too low for convenient access.
OTOH, many people just top up through the valve stem and that works fine. Next time I see a stan's person I'll ask. They have a location here in town so they sometimes drop by the shop to drink our beer.