A process for separating components of a gas mixture using gas-separation copolymer membranes. These membranes use a selective layer made from copolymers of an amorphous perfluorinated dioxolane and a fluorovinyl monomer. The resulting membranes have superior selectivity performance for gas pairs of interest while maintaining fast gas permeance compared to membranes prepared using conventional perfluoropolymers such as Teflon® AF, Hlyflon® AD, and Cytop®.