Overpartitions and Kaur, Rana, and Eyyunni's mex sequences

Kaur, Rana, and Eyyunni recently defined the mex sequence of a partition and established, by analytic methods, connections to two disparate types of partition-related objects. We make a bijection between partitions with certain mex sequences and a uniform family of overpartitions which allows us to provide combinatorial proofs of their results, as they requested.