We study the $E_2$ -algebra $\Lambda \mathfrak {M}_{*,1}:= \coprod _{g\geqslant 0}\Lambda \mathfrak {M}_{g,1}$ consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion $\Omega B\Lambda \mathfrak {M}_{*,1}$ : it is the product of $\Omega ^{\infty }\mathbf {MTSO}(2)$ with a certain free $\Omega ^{\infty }$ -space depending on the family of all boundary-irreducible mapping classes in all mapping class groups $\Gamma _{g,n}$ with $g\geqslant 0$ and $n\geqslant 1$ .

We introduce a general definition for coloured cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from coloured cyclic operads to coloured operads has both adjoints, each of which is relatively simple. Explicit formulae for these adjoints allow us to lift the Cisinski–Moerdijk model structure on the category of coloured operads enriched in simplicial sets to the category of coloured cyclic operads enriched in simplicial sets.