CMOS Timing specifications
The arrow shows the current flowing through the channel of the nMOS element of the left gate, as that transistor turns on due to a rising Vin voltage, as the Cw distributed capacitance discharges.
On the right, Cw is connected to the gate of the MOSFETs, which are insulated from the channels. There is no path for current to flow to ground through the right, aside from a very small transient associated with the gate capacitance of those MOSFETs. A more comprehensive model could show a small current charging those gate capacitances, but the majority of the stored charge in Cw is moved via the channels of the MOSFETs on the left.