So how exactly would I check for a nonlinearity? Would I just get another scope and look at the ramp waveform? Wouldn't the capacitance of the leads affect the switching time of the transistor? The other thing I thought of is putting a known waveform on the input of the fixed scope and seeing if the timebase looks irregular (if the waveform looks stretched or compressed at specific points on the screen).
The later is how it is done. At 50ns/div, a 10 MHz sine wave will have a zero crossing at every division. With x10 magnification, this becomes 5ns/div so a 100 MHz sine wave is needed which gets tricky because the 60 MHz 2213A may not reliably trigger which is why time-mark generators have low frequency trigger outputs.
If there is a problem, then it will show up at the beginning of the 50ns/div sweep.