We demonstrate the existence of steady low-temperature multistage diffusion flames in the counterflow configuration and examine their character as well as that of premixed flames. The character of both diffusion and premixed multistage flames is such that intermediates are produced and consumed in separate oxidation layers. For hydrocarbon fuels in thermodynamic conditions of practical interest, there exist three distinct temperature-dependent classes of reactions that dictate three different oxidation stages, the first being low-temperature peroxy chemistry, the second being intermediate-temperature chemistry, and the third being high-temperature small radical chemistry. For ethers of practical interest, such as diethyl and dibutyl ether, the intermediate-temperature stage is pronounced owing to the additional contribution of the intermediates produced from the scission of the fuel radical, along with the intermediates produced from the decomposition of ketohydroperoxide. For both non-premixed and premixed flames, multistage flames exist for intermediate Damköhler numbers, limited by the classical high-temperature hot flame in the limit of large Damköhler number and a single-stage low-temperature cool flame in the limit of small Damköhler number approaching that of extinction. An abrupt transition between the two-stage flame and single-stage hot flame occurs, whereas a smooth transition between both flames occur for rich premixed flames, for the conditions considered. Premixed and non-premixed multistage flames are particularly relevant to combustion in microgravity and highly turbulent environments, as a result of the long residence times that are possible.