The classical theory of formal models of rigid analytic spaces due to Raynaud introduces the category of admissible R-formal schemes for $R$ a discretely valued ring, which includes locally topologically of finite type formal R-schemes and so-called special formal R-schemes (covered by formal spectra $\mathrm{Spf} R[[x_1, \ldots, x_n]]/(a_1, \ldots, a_n)$). This category is localised with respect to blowups in the special fibre and a "generic fibre" functor $\eta$ defines an equivalence of categories between the localised category of admissible formal $R$-schemes and the category of rigid analytic spaces over $K$.

It is natural to extend the formal scheme part of the construction to
formal schemes over more general base spaces, which was done by
Bosch and Luetkebohmert. The resulting generalised category is called by him "relative
rigid spaces", and the base space $S$ can be, in particular, an
admissible formal $R$-scheme. While the base scheme has a generic
fibre in the sense of Raynaud, I couldn't, to my surprise, find a
construction of generic fibre functor for *relative rigid spaces*
over $S$. I wonder if I overlooked a well-known fact, or if there is an serious obstacle to such construction. I expect that this fibre functor ought to give rigid analytic $K$-spaces fibred over an analytic space closely related to generic fibre of $S$.

For example, the universal Tate curve is naturally regarded as a relative rigid analytic space over $\mathrm{Spf}R[[q]]$, so the generic fibre of it should give rise to the rigid analytic universal family of totally degenerate elliptic curves over the punctured open disc (punctured is important, as the period $q$ cannot vanish).

My question is, therefore: has the construction of such generic fibre functor of relative rigid analytic spaces over admissible formal $R$-schemes been worked out in the literature?