How are sampling frequency of an ADC and switching speed of a multiplexer synchronized?
That is totally up to you, you have to somehow control and synchronize the tasks of sampling and multiplexing.  

You could use a µC, that triggers each conversion and switches the MUX to the next position after each finished conversion. OR you could measure multiple data points at one MUX position and only then  switch the input of the MUX. It completly depends on your application. 

Let's assume two use cases:

1) You sample different audio channels, only one at a time:  You sample with a typical audio sampling frequency of 44,1 kHz. You want to do this for a few seconds or minutes. After that you switch to the next audio input. If you double the sampling frequency you would probably not change the MUX speed, because these two parameters are independent (sampling frequency for your signal quality, MUX frequency for the time slots each input has).  

2) You want to simultaneously capture different channels, e.g. in an oscilloscope: You always take one sample and immediately switch to the next input channel. Doubling the sampling frequency would require you to also double the MUX frequency, because both frequencies are synchronized.