During the past fifteen years, data exchange has been explored in depth and in a variety of different settings. Even though temporal databases constitute a mature area of research studied over several decades, the investigation of temporal data exchange was initiated only very recently. We analyze the properties of universal solutions in temporal data exchange with emphasis on the relationship between universal solutions in the context of concrete time and universal solutions in the context of abstract time. We show that challenges arise even in the setting in which the data exchange specifications involve a single temporal variable. After this, we identify settings, including data exchange settings that involve multiple temporal variables, in which these challenges can be overcome.