We consider the influence of a shifting environment and an advection on the spreading of an invasive species through a model given by the diffusive logistic equation with a free boundary. When the environment is shifting and without advection ( $$\beta =0$$ β = 0 ), Du et al. (Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary. arXiv:1508.06246 , 2015) showed that the species always dies out when the shifting speed $$c_*\ge \mathcal {C}$$ c ∗ ≥ C , and the long-time behavior of the species is determined by trichotomy when the shifting speed $$c_*\in (0,\mathcal {C})$$ c ∗ ∈ ( 0 , C ) . Here we mainly consider the problems with advection and shifting speed $$c_*\in (0,\mathcal {C})$$ c ∗ ∈ ( 0 , C ) (the case $$c_*\ge \mathcal {C}$$ c ∗ ≥ C can be studied by similar methods in this paper). We prove that there exist $$\beta ^*<0$$ β ∗ < 0 and $$\beta _*>0$$ β ∗ > 0 such that the species always dies out in the long-run when $$\beta \le \beta ^*$$ β ≤ β ∗ , while for $$\beta \in (\beta ^*,\beta _*)$$ β ∈ ( β ∗ , β ∗ ) or $$\beta =\beta _*$$ β = β ∗ , the long-time behavior of the species is determined by the corresponding trichotomies respectively.