A hydroxypolyamide (HPA) manufactured from 2,2-bis(3-amino-4-hydroxy phenyl)-hexafluoropropane (APAF) diamine and 5′-terbutyl-m-terphenyl-4,4′′-dicarboxylic acid chloride (tBTpCl), and a copolyimide produced by stochiometric copolymerization of APAF and 4,4′-(hexafluoroisopropylidene) diamine (6FpDA), using the same diacid chloride, were obtained and used as polymeric matrixes in mixed matrix membranes (MMMs) loaded with 20% (w/w) of two porous polymer networks (triptycene-isatin, PPN-1, and triptycene-trifluoroacetophenone, PPN-2). These MMMs, and also the thermally rearranged membranes (TR-MMMs) that underwent a thermal treatment process to convert the o-hydroxypolyamide moieties to polybenzoxazole ones, were characterized, and their gas separation properties evaluated for H2, N2, O2, CH4, and CO2. Both TR process and the addition of PPN increased permeability with minor decreases in selectivity for all gases tested. Excellent results were obtained, in terms of the permeability versus selectivity compromise, for H2/CH4 and H2/N2 separations with membranes approaching the 2008 Robeson’s trade-off line. The best gas separation properties were obtained when PPN-2 was used. Finally, gas permeation was characterized in terms of chain intersegmental distance and fraction of free volume of the membrane along with the kinetic diameters of the permeated gases. The intersegmental distance increased after TR and/or the addition of PPN-2. Permeability followed an exponential dependence with free volume and a quadratic function of the kinetic diameter of the gas.