Multiple-input multiple-output (MIMO) systems employing multicarrier modulation (MCM), including flexible MCM that unifies a number of MCM schemes, have been well studied recently also in their tensor-based formulation. The latter naturally allows for (semi-)blind joint channel estimation/data detection (JCD), of particular interest in future generation massive MIMO systems, given the uniqueness properties of the tensor decomposition (TD) models. Of course, in the presence of time-varying channels, adaptive JCD needs to be implemented. However, the rich literature on recursive TD has only very little been considered in this context and only for orthogonal frequency division multiplexing (OFDM). In this paper, online TD is adapted to realize adaptive JCD for time-varying flexible MIMO-MCM systems. The common assumptions underlying online TD schemes are revisited on this occasion and shown to be inadequate to cope with fast-fading channels. This is demonstrated via simulation results on two of the most popular MCM systems. The results of this work can serve to motivate further study of online TD to extend its applicability in this and other demanding application contexts.