Inspired by graph domination games, various domination-type vertex sequences have been introduced, including the Grundy double dominating sequence (GDDS) of a graph and its associated parameter, the Grundy double domination number (GDDN). The decision version of the problem of computing the GDDN is known to be NP-complete, even when restricted to split graphs and bipartite graphs. In this paper, we establish general tight bounds for the GDDN. We also describe GDDSs for vertex-removed graphs and for the join of two graphs. Applying these results, we prove that computing the GDDN is linear for $P_4$-tidy graphs, thereby solving an open problem previously posed for cographs by B. Brešar et al. in [Brešar, B., Pandey, A., and Sharma, G. (2022). Computational aspects of some vertex sequences of grundy domination-type. Indian J. Discrete Math., 8:21-38].