During the past two decades, 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 first present a chase algorithm for temporal data exchange settings with multiple temporal variables in the context of concrete time and show that this chase algorithm produces universal solutions, provided it does not fail. After this, we examine 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, but also identify cases in which these challenges can be overcome.