Is it a closed system? As in, where does the fluid go when it drips out of
the container?

A liquid like washer fluid will evaporate, probably too fast to be
practical. You'd need something else. Automotive anti-freeze comes to mind
but can be toxic to some animals. If you rigged this thing up to use tubing
to seal up the fluid then you might consider it. Simple 1/4" tubing like
that used for micro-drip irrigation might be right. That and you could put
a 1/4" valve on it to fine-tune the flow amount.