What happens with Source and Drain when I use PWM on the Gate of a MOSFET?
The gate of a MOSFET is basically a capacitive load. The FET's drain current is controlled by the gate to bulk voltage. For discrete FETs the bulk is often connected to the source terminal internally.

This means the problem of driving a FET gate can be modeled as charging a capacitor. So you can archive both cases described in your question.

If the PWM output can drive large currents the FET gate can be charged quickly and  it will turn the drain current on or off with a high slope. This is used for example in switch mode power supplies.

If the PWM output can only drive low currents, the gate capacity together with the line resistance will form a low pass filter. You can increase this effect together with a resistor from PWM output to gate, to implement a very cheap form of digital to analog converter. The duty cycle of the PWM controls the average gate voltage. For this mode you must consider there is a minimum gate voltage (threshold voltage) to archive an effect on the drain.