Trip purpose is essential information supporting tasks in intelligent transportation systems, such as travel behaviour comprehension, location-based service, and urban planning. The observation of trip purpose is a necessary aspect of travel surveys. However, owing to the sampling volume, survey budget, and survey frequency, relying solely on travel surveys in the era of big data is a difficult task. There has long been a demand for an accurate, generalizable, and robust inference method for trip purposes. Although existing studies contributed significant efforts to improve the trip purpose inference, the potential of leveraging the trip chain is insufficient. The spatial correlations and chaining patterns hidden in travelled zones are worthy of further exploration. The unequal importance within trip chains has not been clearly represented. Additionally, complex activity-zone mutual interdependence has not been considered in previous models. Herein, we propose a framework- Dual-Flow Attentive Network with Feature Crossing (DACross), specifically for inferring the chained trip purpose. We form trip chains innovatively that treat trip activities and travelled geographic zones as two chains with mutual interactions. We propose DACross, which consists of two parallel attentive branches and a co-attentive feature crossing module, for fully learning the intra- and inter-chain dependencies. We conducted extensive experiments on four large-scale real-world datasets to evaluate not only the performance of DACross but also the generalizability of the proposed framework among different cities and scenarios. Notably, the Experimental results prove the overall superiority of the proposed DACross.