Disposing of the cores is a matter of personal choice. From a cultural perspective, there may be an advantage to allowing the cores to disintegrate and filter back down into the lawn. Mingling soil and thatch may hasten the natural decomposition of the thatch. The little fluffs of thatch and turf that remain can be collected and composted. Depending on soil type, core disintegration may take a few days to several weeks.
Irrigation helps wash the soil from the cores.
Dragging a piece of cyclone fence or an old metal door mat can speed the process. Running over the cores with a rotary mower can be effective but can dull the blade. Many commercial companies that perform core cultivation break up the cores with a power rake. If the cores are removed from the lawn, compost them before using them as a mulch or soil amendment. Fertilization Nitrogen is the most important nutrient for promoting good turf color and growth.