The present invention utilizes mechanical vapor compression and/or thermal vapor compression integrating compression loops across multiple process stages. A sequential network of compressors is utilized to increase the pressure and condensing temperature of the vapors within each process stage, as intra-vapor flow, and branching between process stages, as inter-vapor flow. Because the vapors available are shared among and between compressor stages, the number of compressors can be reduced, improving economics. Balancing vapor mass flow through incremental compressor stages which traverse multiple process stages by splitting vapors between compressor stages enables the overall vapor-compression system to be tailored to individual process energy requirements and to accommodate dynamic fluctuations in process conditions.