  This is a sequel to our previous articles \cite{Kw23, Kw23a+}. In this work,
we apply recent techniques that fall under the banner of `Period Reciprocity'
to study moments of $GL(3)\times GL(2)$ $L$-functions in the non-archimedean
aspects, with a view towards the `Twisted Moment Conjectures' formulated by
CFKRS.
