Kalman filter for random observation matrix $G_t$
Very interesting question!

I do not know of any reason why you can't do this, but I have had a hard time finding an example.  I finally found this paper, which does an analysis of what happens when the measurements are randomly lost due to transport delays / errors in communication systems.

The main problem that they report is that the error covariance matrix iteration is now stochastic rather than deterministic (as in the usual Kalman filter formulation).