  We consider time-dependent convection-diffusion problems with high P\'eclet
number of order $\mathcal{O}(\varepsilon^{-1})$ in thin three-dimensional
graph-like networks consisting of cylinders that are interconnected by small
domains (nodes) with diameters of order $\mathcal{O}(\varepsilon).$ On the
lateral surfaces of the thin cylinders and the boundaries of the nodes we
account for solution-dependent inhomogeneous Robin boundary conditions which
can render the associated initial-boundary problem to be nonlinear. The
strength of the inhomogeneity is controlled by an intensity factor of order
${\mathcal{O}} (\varepsilon^\alpha)$, $\alpha >0$.
  The asymptotic behaviour of the solution is studied as $\varepsilon \to 0,$
i.e., when the diffusion coefficients are eliminated and the thin
three-diemnsional network is shrunk into a graph. There are three qualitatively
different cases in the asymptotic behaviour of the solution depending on the
value of the intensity parameter $\alpha:$ $\alpha =1,$ $\alpha > 1,$ and
$\alpha \in (0, 1).$ We construct the asymptotic approximation of the solution,
which provides us with the hyperbolic limit model for $\varepsilon \to 0$ for
the first two cases, and prove the corresponding uniform pointwise estimates
and energy estimates. As the main result, we derive uniform pointwise estimates
for the difference between the solutions of the convection-diffusion problem
and the zero-order approximation that includes the solution of the
corresponding hyperbolic limit problem.
