Many characteristics of dwarf carbon stars are broadly consistent with a binary origin, including mass transfer from an evolved companion. While the population overall appears to have old-disc or halo kinematics, roughly 2$,$per cent of these stars exhibit H$alpha$ emission, which in low-mass main-sequence stars is generally associated with rotation and relative youth. Its presence in an older population therefore suggests either irradiation or spin-up. This study presents time-series analyses of photometric and radial-velocity data for seven dwarf carbon stars with H$alpha$ emission. All are shown to have photometric periods in the range 0.2--5.2$,$d, and orbital periods of similar length, consistent with tidal synchronisation. It is hypothesised that dwarf carbon stars with emission lines are the result of close-binary evolution, indicating that low-mass, metal-weak or metal-poor stars can accrete substantial material prior to entering a common-envelope phase.