The conditions for synchronization (equivalently, consensus) in linear and nonlinear switching dynamical systems have been extensively studied. In a previous study, we examined the speed of convergence of linear dynamical systems on switching networks in which each snapshot network defining interaction between dynamical elements is a network Laplacian. We showed that temporal dynamics (i.e., switching) of networks slowed down synchronization processes as compared to the case of aggregate dynamics, i.e., synchronization dynamics occurring on the corresponding static network obtained by the aggregation of the temporal network over time. Here we theoretically extend the results in two ways. First, we derive the conditions imposed on the interaction matrices under which the analytical slowing-down results hold true. The condition turns out to be essentially the same as that for the optimal network, which is known as the condition for the fastest local convergence of nonlinear dynamics on networks. Second, we examine the effect of correlation between different snapshots; in actual temporal networks, the same contact tends to be used consecutively in time. We argue that such temporal correlation further slows down temporal dynamics.