This paper is a generalization of a previous analysis of the effects of a small-amplitude, steady, streamwise vorticity field on the flow over an infinitely thin flat plate in an otherwise uniform stream. That analysis, which is given in Goldstein & Leib (1993), required that the disturbance Reynolds number (i.e. the Reynolds number based on the disturbance velocity and length scale) be infinite while the present paper considers the more general case where this quantity can be finite. The results show how an initially linear perturbation of the upstream flow ultimately leads to a small-amplitude but nonlinear cross-flow far downstream from the leading edge. This flow can, under certain conditions, cause the streamwise velocity profiles to develop distinct shear layers in certain localized spanwise regions. These shear layers, which are remarkably similar to the ones that develop in Tollmien–Schlichting-wave transition (Kovasznay, Komoda & Vasudeva 1962), are highly inflectional and can therefore support the rapidly growing inviscid instabilities that are believed to break down into turbulent spots (Greenspan & Benney 1963, and, subsequently, many others). Numerical computations are carried out for input parameters which approximate the flow conditions of some recent experimental studies of the so-called Klebanoff-mode phenomenon. The results are used to explain some of the experimental observations, and, more importantly, to explain why the averaged quantities usually reported in these experiments do not correlate well with the turbulent-spot formation and therefore with the overall transition process.