We generalize previous results that the lumped kinetics of many symmetric irreversible bimolecular reactions at large times is usually of second order. However, this behavior can be violated in special situations, for which we give conditions. We show the asymptotic second-order rate constant depends on the feed composition in a way that allows only a finite number of possibilities, which can be expressed in terms of the underlying rate constants. Each of these possibilities usually corresponds to a unique asymptotic mixture composition, but a continuum of asymptotic compositions can also exist. The system exhibits a long-term memory when the asymptotic behavior depends on the feed composition. Stability theory is used to develop an algorithm for determining observable asymptotic compositions and rate constants. Numerical examples are provided for binary and ternary mixtures to illustrate some structural features of general bimolecular systems. Implications of the present results for experimental kineticists and process modelers are discussed.