With the expanding share of fluctuating renewable energy, the conventional flexible resources provided by the supply side can no longer deal with the energy imbalance. On the demand side, distributed multi-energy systems with process industry (DMSPI) composed of the district energy supply system (DESS) and the process industry load (PIL) have tremendous potential to provide flexibility for the accommodation of more renewable energy. However, there is little study on the flexibility exploration and characterization of the DMSPI, hindering the utilization of such flexible resources by system operators. To address the problem, this paper proposes a projection-based method to evaluate the integrated flexible region (IFR) of the DMSPI considering the energy conversion in the DESS and production adjustment in the PIL. First, an energy-material integrated model based on the energy hub (EH) is proposed to couple the material flows of the PIL and the energy flows of the DESS. In the proposed model, the production target constraints of the PIL and energy interaction constraints of the DESS are considered. Moreover, the integrated flexibility of the DMSPI is mathematically expressed and defined as the IFR based on the polytopic projection. A calculation method based on vertex enumeration is proposed to find the IFR of the DMSPI. Case studies illustrate the effectiveness of the proposed method to explore the IFR of the DMSPI. The results show that the energy conversion and the adjustable production process endow the DMSPI with a substantial ability to convert and shift its multi-energy demand to provide flexibility for system operation.