  Spectral moment formulae of various shapes have proven to be very successful
in studying the statistics of central $L$-values. In this article, we
establish, in a completely explicit fashion, such formulae for the family of
$GL(3)\times GL(2)$ Rankin-Selberg $L$-functions using the period integral
method. The Kuznetsov and the Voronoi formulae are not needed in our argument.
We also prove the essential analytic properties and explicit formulae for the
integral transform of our moment formulae. It is hoped that our method will
provide insights into moments of $L$-functions for higher-rank groups.
