The inversion of data exchange mappings is one of the thorniest issues in data exchange. In this paper we study inverse data exchange from a novel perspective. Previous work has dealt with the static problem of finding a target-to-source mapping that captures the “inverse” of a source-to-target data exchange mapping. As we will show this approach has some drawbacks when it comes to actually applying the inverse mapping in order to recover a source instance from a materialized target instance. More specifically (1): As is well known, the inverse mappings have to be expressed in a much more powerful language than the mappings they invert. (2): There are simple cases where a source instance computed by the inverse mapping misses sound information that one may easily obtain when the particular target instance is available. (3): In some cases the inverse mapping can introduce unsound information in the recovered source instance. To overcome these drawbacks we focus on the dynamic pro...