Disclosed are systems, methods, devices and kits for sequentially dispensing laundry additive materials into the drum of an automatic washing machine as that machine is used to carry out a fabric laundering operation. To bring about such dispensing of additives, a lidded housing structure is placed within the drum of an automatic washing machine, a multi-compartmented insert, containing different laundry additives in each compartment, is placed within the lidded housing structure, the lid is closed and the machine is run through the several stages of its laundering cycle. Means are provided to open at least one compartment of the insert upon closing of the housing structure lid, thereby dispensing the contents of that opened compartment into the washing machine drum as wash additive materials. Additional means are also provided to open additional compartments of the insert later in the laundering cycle to dispense contents of those additional compartments into the washing machine drum as rinse additive materials. The means for opening such additional compartments of the insert are activated by centrifugal force arising during the spin cycle of the laundering operation.