If you look at 2:10 in this video, you will notice water or something coming from the belly of the challenger 600. Why is this?
It' probably not grey water from the lavatory. You're unlikely to have somebody using the head while taxing out for takeoff. It's condensate from the air conditioning system, not unlike the water drainage you get from a car's air cconditioning evaporator on a hot day.
There are drain lines for the main water separators in the low pressure ducting, as well as drains for the ducts themselves at the lowest point in the underfloor ducting so that water can't collect in the ducts. The cabin ducting drains will use spring check valves so they are only open on the ground and close as soon as the cabin starts to pressurize. Water sloshing along the ducts as the airplane accelerates and brakes will cause the drain flow to start and stop.
In humid weather on the ground, the environmental control system's packs will be running hard in cooling mode, and will produce lots and lots of condensate water. The drains will run to the same heated mast used for the galley and lavatory.
From the Bombardier Challenger 300 Cabin Handbook:
Waste water from the galley and lavatory is drained overboard through valvesand hoses to the forward or aft drain masts. Waste water from the galley driptray and ice drawers is discharged through the forward drain mast. Waste waterfrom the lavatory sink is discharged through the aft drain mast.
Sample image of a drain mast used in a Challenger aircraft:
I believe this image of the bottom of the fuselage shows the drain mast - which corresponds to the source of the dripping water in the video you provided: