Disclosed herein are processes for producing a CO2-depleted product gas stream. The processes involve feeding a natural gas feed stream comprising greater than about 10 vol % CO2 to at least one membrane unit comprising a plurality of polymer membranes to provide a CO2-rich permeate comprising at least 95 vol % CO2 and a CO2-depleted product gas stream. The polymer membranes comprise a crosslinked polyimide polymer having covalent ester crosslinks and have a CO2 permeance of at least 20 GPU and a CO2/CH4 selectivity of greater than 20, at 35 degrees C. and a feed pressure of 100 psia. Also disclosed herein is an apparatus incorporating the crosslinked polyimide polymer for producing a CO2-depleted product gas stream from a natural gas feed stream.