  We present the explicit form of a family of Liouville integrable maps in 3
variables, the so-called triad family of maps and we propose a multi-field
generalisation of the latter. We show that by imposing separability of
variables to the invariants of this family of maps, the $H_{\rm I}$, $H_{\rm
II}$ and $H_{\rm III}^A$ Yang-Baxter maps in general position of singularities
emerge. Two different methods to obtain entwining Yang-Baxter maps are also
presented. The outcomes of the first method are entwining maps associated with
the $H_{\rm I}$, $H_{\rm II}$ and $H_{\rm III}^A$ Yang-Baxter maps, whereas by
the second method we obtain non-periodic entwining maps associated with the
whole $F$ and $H$-list of quadrirational Yang-Baxter maps. Finally, we show how
the transfer maps associated with the $H$-list of Yang-Baxter maps can be
considered as the $(k-1)$-iteration of some maps of simpler form. We refer to
these maps as extended transfer maps and in turn they lead to $k$-point
alternating recurrences which can be considered as alternating versions of some
hierarchies of discrete Painlev\'e equations.
