allows to study certain non-linear problems, especially equations describing physical phenomena with hysteresis effects. A particular case of an operator $\mathcal{M}$ is given by $\mathcal{M}=M_{0}+\partial_{t}^{-1}M_{1}$ for some bounded spatial operators $M_{0},M_{1}.$ The resulting problem then takes the simpler form $$(U,F)\in\partial_{t}M_{0}+M_{1}+\mathcal{A}.$$ Replacing now $M_{0},M_{1}$ by operator-valued multiplication operators, the resulting problem becomes non-autonomous. These problems were studied in [@Picard2013_nonauto] for the case of skew-selfadjoint operators $\mathcal{A}$ and in [@Trostorff2013_nonautoincl] for the case of a maximal monotone relation $\mathcal{A}.$ However, in the case of maximal monotone relations, the authors of [@Trostorff2013_nonautoincl] had to restrict the class of admissible multiplication operators $M_{0}$ and $M_{1}.$ Finally, in [@Waurick2015_nonauto] Waurick proved a well-posedness result for a very abstract class of non-autonomous differential equations, where $M_{0}$ and $M_{1}$ are replaced by arbitrary space-time operators $\mathcal{M},\mathcal{N},$ where $\mathcal{M}$ should have a bounded commutator with $\partial_{t}$. With this result he was able to generalise both the results of [@Picard] and [@Picard2013_nonauto]. However, it does not cover the result for the inclusions. It is the purpose of that article to provide a solution theory for differential inclusions of the form $$(U,F)\in\partial_{t}\mathcal{M}+\mathcal{N}+\mathcal{A},\label{eq:prob}$$ where $\mathcal{M}$ and $\mathcal{N}$ are space-time operators and $\mathcal{A}$ is a maximal monotone relation. We hereby generalise the 