What happens to the flow rate of water through a hydro electric generator when the electrical load changes? This question refers to what happens in the absence of human intervention.
Since the grid is electrically much "larger" than any individual generator, the generator will continue to run at the same speed. What changes is the phase angle between voltage and current, and the magnitude of the current. There can never be more real power out of the generator than what is coming in via the water flow. In the extreme (wouldn't happen in any reasonable normal operation), the grid drives the generator so that the turbine starts pumping the water instead of the water moving the turbine.
There are also control systems around hydro generators at several levels. The fastest response is by controlling the field current. Then there is a slower control system that adjusts valves for the optimal water flow. These systems largely operate autonomously already (once enabled and configured by the humans), so until something drastic happened, a hydro plant would continue working properly as load changed without humans tweaking knobs.
The load effectively change in a typical 24 hours cycle. And, as you probably expected, this change the phase/frequency.
Just connect a speaker to the electric outlet (with a limiting resistor) or look with a scope. You can hear the frequency change about once every 10 minutes. The scope show the change is relatively fast... a few seconds. Then, the new frequency is quite stable until the next 10 minutes time window where the frequency is adjusted again. This adjustment is made mostly to help the clocks to keep correct time. In the recent past, most clocks were using a synchronous motor which was rotating at a multiple of 60 Hz, such as 10 turns per seconds for 6 poles models. The frequency would be adjusted one time slightly above 60 Hz, then slightly below. The total count was the important factor, so, if a previous correction overshooted too much, it could happen that the frequency is set the other side of the 60 Hz goal for 20 minutes or more. In am pretty sure that thermal central also need to adjust the frequency at some time interval.