A process for separating components of a gas mixture using gas-separation copolymer membranes. These membranes use a selective layer made from copolymers of partially fluorinated or perfluorinated dioxolane monomers and a second monomer, such as dioxane or a partially fluorinated dioxolane. 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, Hyflon® AD, and Cytop®.