We study the asymptotic stability of traveling fronts and the front's velocity selection problem for the time-delayed monostable equation with Lipschitz continuous reaction term . We also assume that g is -smooth in some neighbourhood of the equilibria 0 and to . In difference with the previous works, we do not impose any convexity or subtangency condition on the graph of g so that equation can possess the pushed traveling fronts. Our first main result says that the non-critical wavefronts of with monotone g are globally nonlinearly stable. In the special and easier case when the Lipschitz constant for g coincides with , we prove a series of results concerning the exponential (asymptotic) stability of non-critical (respectively, critical) fronts for the monostable model . As an application, we present a criterion of the absolute global stability of non-critical wavefronts to the diffusive non-monotone Nicholson's blowflies equation.