  We consider the general form of "Correlated Worldline" (CWL) theories of
quantum gravity. We show that one can have 2 different kinds of CWL theory, in
which the generating functional is written as either a sum or a product over
multiple copies of the coupled matter and gravitational fields. In both
versions, the paths in a functional formulation are correlated via gravity
itself, causing a breakdown of the superposition principle; however, the
product form survives consistency tests not satisfied by the summed form. To
better understand the structure of these two theories, we show how to perform
diagrammatic expansions in the gravitational coupling for each version of CWL
theory, using particle propagation and scalar fields as examples. We explicitly
calculate contributions to 2-point and 4-point functions, again for each
version of the theory, up to 2nd-order in the gravitational coupling.
