Realize that comparators are just amplifiers without gain compensation. And, that amplifiers are just analog filters: they have a gain(frequency) response.
Suppose you biased the inputs, such that the output sits halfway between logic levels. (This might be very tricky indeed if the comparator has built-in hysteresis, but let's suppose it doesn't. Let's also suppose it's noiseless, too...) If you apply a very small step now (perhaps several uV), the output will at first do nothing, then after some real time delay (which might be fractional ns), it begins moving. Some time later (10s of ns up to us), the output ramps up, then settles to a new value (which is still near halfway, but different by a fraction of a volt -- if the DC open loop gain is 100dB, then 1uV input change makes 0.1V output change).
Now suppose the input is increased to a few mV. The output would have to jump several volts, maybe a thousand volts, if it could! But it can't, because it's constrained by the supply voltages. Not only that, but internal nodes are also constrained, so you get slew rate limiting, a minimum propagation delay, and a minimum output rise/fall time.
Think of it this way: what you see at the output is a many-times magnification of a very small segment of rise/fall. The comparator's open loop time constant might be microseconds, but because it's only linear over a tiny fractional-mV segment of its full input range, the output gets chopped into "1" and "0" with a very short time spent going between them. And, because of internal propagation and slew-rate limiting, it's actually quite a bit slower than a linear magnification: if it were fully linear (up to the output clipping), the propagation delay should remain inversely proportional to input level. But it's not, it saturates to a minimum delay, because other internal components do the same limiting function.
In a sense, you can model a real comparator as a chain of simple comparators, which aren't really comparators themselves as such, but are amplifiers with a modest linear range and an output range that saturates between "high" and "low" limits.
Given this model, you can evaluate what will happen, in your head!
For zero overshoot (i.e., the input doesn't reverse polarity at all), the output might eventually stabilize around mid-supply (i.e., the linear condition I talked about first). But with zero input difference, it will take a very long time indeed to get there (on the order of (DC gain) / GBW seconds), so nothing really happens, not over a time span of microseconds.
For a little overshoot, some mV perhaps, and 20ns duration of that, you still won't see an output change, because that will be enough to move the first stage, maybe, but the internal voltages don't cross the linear range of the other stages.
For a large overshoot, 10s or 100s of mV, for 20ns, enough will propagate along the chain of internal stages, that you see an output. The output itself may simply twitch, or it may cross whatever logic threshold you are using (remember that digital is a matter of definition: everything is analog, first and foremost!). The rise/fall time of that twitch may not be very fast, because the input wasn't very strong.
For a large, sustained overshoot, like 100mV for longer than t_PLH duration, the output rises as fast as it can, because the first internal stage is driven into saturation almost immediately, which saturates the next stage, and so on. And the output stays there, until enough input is given, in the reverse direction, to do all this again.
Any amplifier that has a constant GBW behavior (which is typical of the average op-amp and comparator) is best modeled as an integrator with saturation: you apply an input, and this causes the output to
change. The output doesn't simply move to a new, proportional voltage. Indeed, the voltage change is very nearly proportional to the input volts
*time applied.
What else looks like this? An inductor draws current, proportional to the volt*seconds applied. So the input of an inverting op-amp with negative feedback looks like an inductor! And, the same stimulus (a bump of voltage for some duration) is necessary to get a comparator to respond, too!
Tim