Implications of a problem being in XP when parameterized by diameter
I think Figure 1 (page 4) of the paper "New Races in Parameterized Algorithmics" of Komusiewicz and Niedermeier is what you are looking for.

In particular, being in XP for the parameter diameter implies being in XP for parameters: min dominating set, max independent set, minimum clique cover, distance to cograph, distance to co-cluster, distance to clique, distance to cluster, vertex cover, and cluster editing.