A domain $R$ is \emph{perinormal} if every going-down overring is flat and a perinormal domain $R$ is \emph{globally perinormal} if every flat overring is a localization of $R$ [Epstein-Shapiro 2016]. I show that global perinormality is preserved in a pullback construction which encompasses a classical $D+M$ construction. In doing so, a result is given for the transfer of the property that every flat overring is a localization in the pullback construction considered.

Revised proof of Proposition 3.3, replaced Lemma 2.2 which was missing an assumption with Observation 2.2 which is all that is necessary, added Lemma 3.2. Other minor revisions made. (7 pages)