It would at a technical level need to be serialisable which I don't think a class would be. Think of JSON, assume everything is effectively sent in that format. So internally the engine might be serialising and de-serialising basic types like numbers and strings.
I mean I guess just about anything could be sent, as long as you could reconstruct the class on the other end. Like there's a separate implementation of the same class, and you turn it into binary. All references would be lost.
Well now I'm talking about things I don't really know. How complicated is the data you want to send?