The propagation delay is simply "the amount of time it takes for the head of the signal to travel from the sender to the receiver" (from [here][1]here), it is not the bandwidth, which is how fast the system can respond to the input.
The output will behave as expected, but delayed by the propagation delay time.
With different rise and fall propagation delays it will be more complicated, because different part of the signal will be delayed differently, and the behaviour will not be so obvious.
Of course, all of this is valid for simple systems (i.e. the inverter), increasing the complexity will lead to incresed chance of unpredictable behaviour. (i.e. the [race condition][2]race condition)
There are even circuits based on the propagation delay, i.e. the [ring oscillator][3]
[1]: https://en.wikipedia.org/wiki/Propagation_delay
[2]: https://en.wikipedia.org/wiki/Race_condition
[3]:https://en.wikipedia.org/wiki/Ring_oscillatorring oscillator