We investigate in this paper a scalar reaction diffusion equation with a nonlinear reaction term depending on x-ct. Here, c is a prescribed parameter modeling the speed of a climate change and we wonder whether a population will survive or not, that is, we want to determine the large-time behavior of the associated solution. This problem has been solved recently when the nonlinearity is of KPP type. We consider in the present paper general reaction terms, that are only assumed to be negative at infinity. Using a new variational approach, we construct two thresholds determining the existence and the non-existence of traveling waves. Numerics support the conjecture that the two thresholds are equal. We then prove that any solution of the initial-value problem converges at large times, either to 0 or to a travelling wave. In the case of bistable nonlinearities, where the steady state 0 is assumed to be stable, our results lead to constrasting phenomena with respect to the KPP framework. Lastly, we illustrate our results and discuss several open questions through numerics.