The CUPID-Mo experiment demonstrated cryogenic scintillating Li$_2^{100}$MoO$_4$ calorimeters for the use in neutrinoless double beta decay search (0νββ) in CUPID (CUORE Upgrade with Particle IDentification). It consisted of an array of twenty detector modules with ~200-g detectors and Ge wafer based light detectors enabling particle identification. It was operated at the Laboratoire Souterrain de Modane (France) from 2018 to 2020. In this work, we demonstrate the applicability of the experimental design towards a search for two-neutrino and neutrinoless double beta decays of Mo-100 to the 1st $0_1^+$ and $2_1^+$ excited states of Ru-100 using the full CUPID-Mo exposure (2.71 kg-yr of Li$_2^{100}$MoO$_4$). We present results for the measured half-life of the 2νββ decay to the $0_1^+$ state. The acquired statistics allows us to make the first comparison of the single state (SSD) and higher state (HSD) 2νββ decay models for the decay to the $0_1^+$ excited state of Ru-100 demonstrating the potential of the technique for future experiments. We also present the analysis of the neutrinoless decay mode to the $2_1^+$ and the $0_1^+$ states. These results are expected to be the most stringent constraints on these processes in Mo-100 to date.