  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.
