When extending inferences from a randomized trial to a new target population, the transportability condition for conditional difference effect measures is invoked to identify the marginal causal mean difference in the target population. However, many clinical investigators believe that conditional relative effect measures are more likely to be "transportable" between populations. Here, we examine the identification and estimation of the marginal counterfactual mean difference and ratio under the transportability condition for conditional relative effect measures. We obtain identification results for two scenarios that often arise in practice when individuals in the target population (1) only have access to the control treatment, and (2) have access to the control and other treatments but not necessarily the experimental treatment evaluated in the trial. We then propose model and rate multiply robust and nonparametric efficient estimators that allow for the use of data-adaptive methods to model the nuisance functions. We examine the performance of the methods in simulation studies and illustrate their use with data from two trials of paliperidone for patients with schizophrenia. We conclude that the proposed methods are attractive when background knowledge suggests that the transportability condition for conditional relative effect measures is more plausible than alternative conditions.