  In \cite{Bapic, Tang, Zheng} a new method for the secondary construction of
vectorial/Boolean bent functions via the so-called $(P_U)$ property was
introduced. In 2018, Qi et al. generalized the methods in \cite{Tang} for the
construction of $p$-ary weakly regular bent functions. The objective of this
paper is to further generalize these constructions, following the ideas in
\cite{Bapic, Zheng}, for secondary constructions of vectorial $p$-ary weakly
regular bent and plateaued functions. We also present some infinite families of
such functions via the $p$-ary Maiorana-McFarland class. Additionally, we give
another characterization of the $(P_U)$ property for the $p$-ary case via
second-order derivatives, as it was done for the Boolean case in \cite{Zheng}.
