The reason why the output waveform of gate voltage boost is not almost the straight line?
Weak FET Conduction (leakage) resistance may cause some exponential decay. 
Thus rising edge charge pump is higher by 370~372mV and  falling edge is less after each stage.  Each stage adds near the same amount to the previous.

Speculation

Increasing the clock rate will reduce the slope but at some point will reduce the step rise due to C ratios and Conduction ratios.