  We consider for a small parameter $\varepsilon >0$ a parabolic
convection-diffusion problem with P\'eclet number of order
$\mathcal{O}(\varepsilon^{-1})$ in a three-dimensional graph-like junction
consisting of thin curvilinear cylinders with radii of order
$\mathcal{O}(\varepsilon)$ connected through a domain (node) of diameter
$\mathcal{O}(\varepsilon).$ Inhomogeneous Neumann type boundary conditions,
that involve convective and diffusive fluxes, are prescribed both on the
lateral surfaces of the thin cylinders and the boundary of the node.
  The asymptotic behaviour of the solution is studied as $\varepsilon \to 0,$
i.e., when the diffusion coefficients are eliminated and the thin junction is
shrunk into a three-part graph connected in a single vertex. A rigorous
procedure for the construction of the complete asymptotic expansion of the
solution is developed and the corresponding energetic and uniform pointwise
estimates are proven. Depending on the directions of the limit convective
fluxes, the corresponding limit problems $(\varepsilon = 0)$ are derived in the
form of first-order hyperbolic differential equations on the one-dimensional
branches with novel gluing conditions at the vertex. These generalize the
classical Kirchhoff transmission conditions and might require the solution of a
three-dimensional cell-like problem associated with the vertex to account for
the local geometric inhomogeneity of the node and the physical processes in the
node. The asymptotic ansatz consists of three parts, namely, the regular part,
node-layer part, and boundary-layer one. Their coefficients are classical
solutions to mixed-dimensional limit problems. The existence and other
properties of those solutions are analyzed.

Analysis of PDEs