In a recent publication, we established a close relationship between light-curve asymmetries in Mira variables and indicators of their dust mass-loss rate. The light-curve asymmetries appear to be related to the stars' third dredge-up (3DUP) activity. We aim to reveal the evolutionary status of M-type Miras with light-curve asymmetries ("bumps") within the spectral sequence M-S-C, determine their mass-loss properties, and check possible evolutionary scenarios. We analysed a sample of 3100 Miras collected from the ASAS database, distinguishing between symmetric and asymmetric light curves. We determined their periods, luminosity functions, and period-luminosity relations, their locations relative to the Galactic midplane, as well as mass-loss rate indicators through the 2MASS-WISE colours and the Gaia-2MASS diagram. The M-type Miras with symmetric light curves are generally found to have shorter periods, lower luminosities, a larger average distance to the Galactic midplane, and lower initial mass than the M-type Miras with asymmetries. In addition, 25 Miras are proposed as candidates for new carbon stars. We propose that the M-type Miras have two distinct populations: M-type Miras with symmetric light curves, which have lower initial mass progenitors than M-type Miras with asymmetries, which show signs of 3DUP activity and are the link to the S-type Miras.