The 2D equations in the Kirchhoff-Love theory are subjected to asymptotic analysis in the case of free interfacial vibrations of a longitudinally inhomogeneous infinite cylindrical shell. Three types of interfacial vibrations, associated with bending, super low-frequency semi-membrane, and extensional motions, are investigated. It is remarkable that for extensional modes natural frequencies have asymptotically small imaginary parts caused by a weak coupling with propagating bending waves. Bending and extensional vibrations correspond to Stonely-type plate waves, while semi-membrane ones are strongly dependent on shell curvature and do not allow flat plate interpretation. The paper represents generalization of the recent authors' publication [Kaplunov et al., J. Acoust. Soc. Am. 107, 1383-1393 (2000)] dealing with edge vibrations of a semi-infinite cylindrical shell.