We study the behavior of a generalized consensus dynamics on a temporal network of interactions, the activity-driven network with attractiveness. In this temporal network model, agents are endowed with an intrinsic activity a, ruling the rate at which they generate connections, and an intrinsic attractiveness b, modulating the rate at which they receive connections. The consensus dynamics considered is a mixed voter and Moran dynamics. Each agent, either in state 0 or 1, modifies his or her state when connecting with a peer. Thus, an active agent copies his or her state from the peer (with probability p) or imposes his or her state to him or her (with the complementary probability 1-p). Applying a heterogeneous mean-field approach, we derive a differential equation for the average density of voters with activity a and attractiveness b in state 1, which we use to evaluate the average time to reach consensus and the exit probability, defined as the probability that a single agent with activity a and attractiveness b eventually imposes his or her state to a pool of initially unanimous population in the opposite state. We study a number of particular cases, finding an excellent agreement with numerical simulations of the model. Interestingly, we observe a symmetry between voter and Moran dynamics in pure activity-driven networks and their static integrated counterparts that exemplifies the strong differences that a time-varying network can impose on dynamical processes.