What are these droppings on the core flute?

Hi Rocko, this is chalkbrood or stonebrood, its a fungus infecting the larva stage of your brood. You should do an inspection of your hive as soon as you can to make an assessment on the severity. If you have other hives in the vicinity then be sure not to spread the fungus with your hive tools or gloves. The dark and whiteness you see on the brood mummies are the fruiting spores of the fungus, be extra vigilant not to spread these around your yard as the spores remain viable for decades. See the post in the following link below for a few tips.