Your Brother machine is designed to stop ALL printing operations when any of the ink cartridges are empty. This is to ensure the life of the print head and maintain premium print quality. If the unit were to continue printing, print jobs and cleaning cycles would suck air from the empty cartridge and damage the print head. Even if you print in black only or change the print settings to "Mono" or "Grayscale" to avoid printing in color, the programmed cleaning cycles would still suck air from the empty cartridge, because they are programmed to clean ALL the ink nozzles. The only way to continue printing is to replace the empty ink cartridge(s).
For FAX/MFC models with Fax function:
While memory is available, black and white faxes will continue to be received into memory. If a color fax is being received, the Brother machine's "handshake" will request the fax be sent as black and white. If the sending machine has the ability to convert it, the color fax will be received into memory as a black and white fax. All faxes received into memory will automatically print once the ink cartridge is replaced.