Movable antenna (MA) introduces a new degree of freedom for future wireless communication systems by enabling the adaptive adjustment of antenna positions. Its large-range movement renders wireless channels transmission into the near-field region, which brings new performance enhancement for integrated sensing and communication (ISAC). This paper proposes a novel multi-stage design framework for broadband near-field ISAC assisted by MA. The framework first divides the MA movement area into multiple subregions, and employs the Newtonized orthogonal matching pursuit algorithm (NOMP) to achieve high-precision angle estimation in each subregion. Subsequently, a method called near-field localization via subregion ray clustering (LSRC) is proposed for identifying the positions of scatterers. This method finds the coordinates of each scatterer by jointly processing the angle estimates across all subregions. Finally, according to the estimated locations of the scatterers, the near-field channel estimation (CE) is refined for improving communication performance. Simulation results demonstrate that the proposed scheme can significantly enhance MA sensing accuracy and CE, providing an efficient solution for MA-aided near-field ISAC.