  In line with the notion of probabilistic rough paths introduced in the
previous contribution \cite{salkeld2021Probabilistic}, we address corresponding
random controlled rough paths (first introduced in
\cite{2019arXiv180205882.2B}), the structure of which is indexed by Lions
forests. These are statistical distributions over the space of paths described
by the combination of a jet on the underlying probabilistic rough path and a
remainder term. The regularity of the latter facilitates the definition of a
rough integral.
  We establish closedness and stability of two key operators on random
controlled rough paths: rough integration and composition by a smooth function
on the Wasserstein space. These are important results towards a complete theory
of rough McKean-Vlasov equations that is still in gestation. The proof goes
through a higher-order Taylor expansion for the Lions derivative which we
rigorously expound.
  The coupled Hopf algebra structure (see \cite{salkeld2021Probabilistic}) and
the Lions-Taylor expansion (established in Section
\ref{section:TaylorExpansions}) introduce a number of additional challenges
which mean these results are not simply a natural extension of classical
theory. We dedicate this work to pursuing these details.
