Is there any keyboard shortcut to switch my mouse cursor from one desktop to another?
There's no shortcut to my knowledge but if you are willing to get your hands a little dirty you can easily achieve this with a combination of 3rd party software, here is how I'd go about it:


Download and install MouseTools.
Download and install Keyboard Maestro.
Add KM Macro based on keys combination (to switch from one screen to another, one Macro for each display).
Finally for each macro add the Shell action with MouseTools -x xValue -y yValue (which are the coordinate on the display where you want the mouse to land).