In this paper, we consider a free boundary model in one space dimension which describes the spreading of a species subject to climate change, where favorable environment is shifting away with a constant speed c > 0 and replaced by a deteriorated yet still favorable environment. We obtain two threshold speeds c1 < c0 and a complete classification of the long-time dynamics of the model, which reveals significant differences between the cases 0 < c < c1, c = c1, c1 < c < c0 and c ≥ c0. For example, when c1 < c < c0, for a suitably parameterized family of initial functions uσ0 increasing continuously in σ, we show that there exists 0 < σ⁎ < σ⁎ < ∞ such that the species vanishes eventually when σ ∈ (0, σ⁎], it spreads with asymptotic speed c1 when σ ∈ (σ⁎, σ⁎), it spreads with forced speed c when σ = σ⁎, and it spreads with speed c0 when σ > σ⁎. Moreover, in the last case, while the spreading front propagates with asymptotic speed c0, the profile of the population density function u(t, x) approaches a propagating pair consisting of a traveling wave with speed c and a semi-wave with speed c0.