We have determined the rate constants and branching ratios for the reactions of CO+ with CH4 and CD4 in a variable-temperature selected ion flow tube. We find that the rate constants are collisional for all temperatures measured (193-700 K for CH4 and 193-500 K for CD4). For the CH4 reaction, three product channels are identified, which include charge transfer (CH4+ + CO), H-atom transfer (HCO+ + CH3), and H-atom expulsion (CH3CO+ + H). H-atom transfer is slightly preferred to charge transfer at low temperature, with the charge-transfer product increasing in contribution as the temperature is increased (H-atom expulsion is a minor product for all temperatures). Analogous products are identified for the CD4 reaction. Density functional calculations on the CO+ + CH4 reaction were also conducted, revealing that the relative temperature dependences of the charge-transfer and H-atom transfer pathways are consistent with an initial charge transfer followed by proton transfer.