  This paper concerns the nonautonomous reaction-diffusion equation \[
u_t=u_{xx}+ug(t,x-ct,u), \quad t>0,x\in\mathbb{R}, \] where $c\in\mathbb{R}$ is
the shifting speed, and the time periodic nonlinearity $ug(t,\xi,u)$ is
asymptotically of KPP type as $\xi \to-\infty$ and is negative as
$\xi\to+\infty$. Under a subhomogeneity condition, we show that there is
$c^*>0$ such that a unique forced time periodic wave exists if and only $|c|<
c^*$ and it attracts other solutions in a certain sense according to the tail
behavior of initial values. In the case where $|c|\ge c^*$, the propagation
dynamics resembles that of the limiting system as $\xi\to\pm \infty$, depending
on the shifting direction.
