Log‐concavity of multivariate distributions is an important concept in general and has a very special place in the field of Reliability Theory. An attempt has been made in this paper to study preservation results for (i) the discrete version of multivariate log‐concavity for multistate series and multistate parallel systems consisting of n$$ n $$ independent components, states of both components and systems being represented by elements in a subset S2={0,1,2}$$ {S}_2=\left\{0,1,2\right\} $$ of SM={0,1,2,…,M},$$ {S}_M=\left\{0,1,2,\dots, M\right\}, $$ and (ii) the continuous version of multivariate log‐concavity under multistate series and multistate parallel systems made up of n$$ n $$ independent components and states of both, systems and components, taking values in the set SM$$ {S}_M $$. These results for discrete and continuous versions of log‐concavity have also been extended to systems that are formed using both multistate series and multistate‐parallel systems. Further, the results in (ii) have been used to obtain important and useful bounds on joint probabilities related to times spent by multistate components, multistate series, multistate parallel systems, and the combinations thereof.