  We study some permanence properties of the relative $F$-rational signature
defined and studied by Smirnov--Tucker. We show that this invariant is
compatible with the gamma-construction, and then derive other main results from
the $F$-finite case established by Smirnov--Tucker. We also obtain limited
results about the $F$-rational signature defined and studied by Hochster--Yao.
We explore some features of the gamma-construction along the way, which may be
of independent interest.
