Conductance of inhomogeneous Luttinger liquids with a finite bandwidth

•Tunneling conductance of inhomogeneous Luttinger liquids with a finite bandwidth.•Generalization of the results of Matveev et al. (PRL 71 (20) 1993) to arbitrary strength of interactions.•Novel physics seen in the form of a weakly temperature dependent conductance for strong repulsions.•Unusual high conductance observed for a weak barrier and strongly repulsive mutual interactions at low temperatures.