The present invention involves the use of a multi-stage membrane system for gas, vapor, and liquid separations. In this multi-stage membrane system, high selectivity and high permeance or at least high selectivity polybenzoxazole membranes or cross-linked polybenzoxazole membranes are applied for a pre-membrane or both the pre-membrane and the secondary membrane. A primary membrane can be from conventional glassy polymers. This multi-stage membrane system can reduce inter-stage compression cost, increase product recovery and product purity for gas, vapor, and liquid separations. It can also save the cost compared to the system using all the high cost polybenzoxazole membranes or cross-linked polybenzoxazole membranes.