NOVELTY - Producing a higher hydrocarbon (C2+) composition, comprises during a first time interval, (i) passing oxidant through a flow-through reactor to transfer oxygen to oxygen storage material, (ii) storing a portion of the transferred oxygen with the material and (iii) lessening the passing of the oxidant via the reactor, and during a second time interval, (iA) passing a hydrocarbon reactant via the flow-through reactor, and (iiA) releasing a portion of the stored oxygen and reacting a portion of the released oxygen with a portion of the reactant in the presence of hydrocarbon conversion catalyst.    USE - The method is useful for producing a C2+ composition including ethane, ethylene (all claimed) and/or propylene.    ADVANTAGE - The method produces the C2+ composition with reduced energy consumption, reduction in undesired combustion reactions, and high selectivity.    DETAILED DESCRIPTION - Producing a hydrocarbon (C2+) composition, comprises either (a) providing a flow-through reactor comprising a hydrocarbon conversion catalyst and an oxygen storage material, (b) during a first time interval, (i) passing oxidant through the flow-through reactor to transfer oxygen from the oxidant to the oxygen storage material, (ii) storing at least a portion of the transferred oxygen with the oxygen storage material, and (iii) lessening or discontinuing the passing of the oxidant through the flow-through reactor, and (c) during a second time interval, (iA) passing a hydrocarbon reactant through the flow-through reactor, and (iiA) releasing at least a portion of the stored oxygen and reacting at least a portion of the released oxygen with at least a portion of the hydrocarbon reactant in the presence of the hydrocarbon conversion catalyst to produce a reaction mixture comprising a C2+ composition, and (d) conducting at least a portion of the reaction mixture away from the flow-through reactor, or (a1) providing a flow-through reactor comprising a thermal mass, a hydrocarbon conversion catalyst, and an oxygen storage material, (b1) during a first time interval (iB) passing a heating fluid comprising an oxidant and a hydrocarbon fuel through the flow-through reactor to produce a combustion gas and transfer heat to the thermal mass, (iiB) transferring oxygen from the oxidant to the oxygen storage material and storing at least a portion of the transferred oxygen with the oxygen storage material, and (iiiB) lessening or discontinuing the passing of the heating fluid through the flow-through reactor, (c1) during a second time interval, passing a sweep fluid through the flow-through reactor to remove at least a portion of the combustion gas, (d1) during a third time interval, (iC) passing a hydrocarbon reactant through the flow-through reactor to contact the heated thermal mass and heat the hydrocarbon reactant, where the hydrocarbon reactant comprises 10 wt.%, and (iiC) releasing at least a portion of the stored oxygen and reacting at least a portion of the heated hydrocarbon reactant's methane with at least a portion of the released oxygen in the presence of the hydrocarbon conversion catalyst produce a reaction mixture comprising a C2+ composition including ethane and/or ethylene produced from the methane, (e1) during a fourth time interval, passing a sweep fluid through the flow-through reactor to remove at least a portion of the reaction mixture from the flow-through reactor. An INDEPENDENT CLAIM is included for the flow-through reactor for producing the C2+ composition, comprising (A) a reactor vessel, (B) first and second apertures adapted for fluid flow into and out of the reactor vessel, (C) an oxygen storage material located in the reactor vessel, and (D) a hydrocarbon conversion catalyst located in the reactor vessel, where the hydrocarbon conversion catalyst has an oxidative coupling functionality and/or oxydehydrogenation functionality, the oxygen storage material and hydrocarbon conversion catalyst are in thermal and diffusive contact, and the first and second apertures are in diffusive contact with oxygen storage material and hydrocarbon conversion catalyst.    DESCRIPTION OF DRAWING(S) - The figure shows a schematic view of a flow-through reactor.    Reaction zone (C)    First thermal mass segments (M1)    Second thermal mass segments. (M2)