Is there an easy way to switch desktops on the remote computer using "back to my mac"
The key command is control and left arrow or right arrow.

You can customize things further in Mission Control preferences as well.

Depending on which VNC/RDC client you are using - the command keys might act locally if you are browsing from OS X - so you might need to eliminate multiple screens to pass those through. 

Alternatively, you can use/set a hot key for mission control and choose which screen you want from the top of the window. Changing the local shortcut will allow it to pass through to the remote OS X if that's blocking you.