  In this paper we study the Grundy domination number on the $X$-join product
$G\hookleftarrow \mathcal R$ of a graph $G$ and a family of graphs $\mathcal
R=\{G_v: v\in V(G)\}$. The results led us to extend the few known families of
graphs where this parameter can be efficiently computed. We prove that if, for
all $v\in V(G)$, the Grundy domination number of $G_v$ is given, and $G$ is a
power of a cycle, a power of a path, or a split graph, computing the Grundy
domination number of $G\hookleftarrow \mathcal R$ can be done in polynomial
time. In particular, the results for power of cycles and paths are derived from
a polynomial reduction to the Maximum Weight Independent Set problem on these
graphs.
  As a consequence, we derive closed formulas to compute the Grundy domination
number of the lexicographic product $G\circ H$ when $G$ is a power of a cycle,
a power of a path or a split graph, generalizing the results on cycles and
paths given by Bresar et al. in 2016. Moreover, the results on the $X$-join
product when $G$ is a split graph also provide polynomial-time algorithms to
compute the Grundy domination number for $(q,q-4)$ graphs, partner limited
graphs and extended $P_4$-laden graphs, graph classes which are high in the
hierarchy of few $P_4$'s graphs.
