MOld growing in ducts occurs when the temperature of the duct sufaces are near or below the dew point of the duct. This can occur in the summer during the off cooling cycle of a/c. The water loaded cooling coil humidifies the air in the ducts until the water on the coil totally evaporates. This can take days or weeks. The solution is daly drying of the ducts and coil. A similar problem is air blown water off the cooling coil into the ducts. If a whole house dehumidifier is used, discharge the dry air into the a/c supply. This slowly dries the ducts and a/c coil avoiding mold growth.
Mold can also grow during the cold weather. When the ducts are located in a cold crawl or attic, the duct temperature is below the dew point of the air inside the home during the off heating cycle. If the ducts are cold and wet for days, expect mold to grow on the surfaces. Operating the fan in the "on" mode would keep the ducts warm and dry during winter. Or avoid cold attic a/c installations in cold climates. TB