Does Flight Following automatically include permission to fly through class C or D airspace?
Lets back up a minute and discuss what is required to enter C and D airspace.
For C and D airspace you must establish 2 way radio communications with tower or approach controller. Let's first discuss how this works if you are not taking Flight Following. As you approach Class C airspace you hail them and wait for a response, for example,

You: &quot;Allentown Approach Piper 1234J @ 3000 en route to Queen City Municipal&quot;
Controller: &quot;1234J Squawk 1-3-5-2 and Ident&quot;

Since the controller responded with your tail number you may enter the Charlie airspace (Delta has the same requirements).
Now when you are under Flight Following, ATC will coordinate a handoff to the Approach or Tower for the airspace you are about to enter, or ATC will say &quot;radar services terminated,&quot; and you must contact the controller/tower yourself before entering.
You SHOULD NOT change frequencies until you are told to do so, since ATC may be trying to give you a traffic advisory. You can however tune your second radio (if you have one) to the approach/tower you are approaching, and begin to monitor traffic.