This paper extends a previous theoretical study (Sabelnikov and Lipatnikov, 2013) of the influence of countergradient transport (CGT) on the speed of a statistically stationary, planar, 1D premixed flame that passively propagates in homogenous turbulence in the form of a traveling wave, i.e. retains its mean thickness and structure. While two particular models of the mean rate of product creation were addressed in the previous article, with the shape of the rate as a function of the Favre-averaged combustion progress variable being concave in both cases, the present paper deals with a more general model that subsumes both concave functions and functions with an inflection point, i.e. a point where the function changes from being concave to convex or vice versa. In this more general case, transition from pulled (flame speed is controlled by processes localized to the flame leading edge) to pushed (flame speed is controlled by processes within the entire flame brush) flames can occur both due to interplay of the nonlinear reaction term and a nonlinear convection term associated with CGT and due to the change of the shape of the reaction term in the absence of CGT. Explicit pushed traveling wave solutions to the studied problem are theoretically derived and conditions under that developing flames approach either pushed or pulled traveling wave solution are obtained by analyzing the governing equations at the flame leading edge and invoking the steepness selection criterion which highlights traveling wave with the steepest profile at the leading edge. Other analytical results include conditions for transition from pulled to pushed premixed turbulent flames, dependence of flame speed on the magnitude of the CGT term and the shape of the mean reaction rate, analytical expressions for the mean thickness of the pushed flames and turbulent scalar flux within the pushed flames. All these theoretical findings are validated by results of unsteady numerical simulations of the initial boundary value problem with steep initial wave profiles. (C) 2015 The Combustion Institute. Published by Elsevier Inc. All rights reserved.