We present a system of reaction-diffusion equations to model population dynamics within climate-driven moving habitats. As the Earth warms, temperature isoclines shift and many species have or will have to shift their geographical ranges to follow their suitable temperature regime. Our mathematical model’s spatial domain is the whole real line, where a bounded interval represents the suitable habitat and the unbounded interval, the unsuitable habitat. The points at the end of the bounded interval are called the interface, and they move in time to represent the shifting temperature isoclines. Unique to our models is a general description of the movement of these interface points and the inclusion of habitat-dependent movement rates and habitat preference. The assumptions on movement rates and habitat preference result in a jump in density across habitat types. We build and numerically validate a finite difference scheme to study the transient dynamics, and, when they exist, steady-state solutions. Our numerical scheme uses a conformal moving mesh and a modification of the finite difference scheme to capture the jumps in density. We apply our scheme to two relevant climate-shifting scenarios: constant nonmatching speeds and accelerating speeds. In the former, we find that a strong preference for the suitable habitat helps the population persist at faster shifting speeds, yet sustains a smaller total population at slower shifting speeds. In the latter, we find that a high preference for the suitable habitat at lower acceleration rates increases the time to extinction but the opposite is true at faster acceleration rates.