Speaking of the chamber housing the sacred mantle of the Prophet:

Under Sultan Ahmed (III) visiting the mantle became an elaborate state ceremony that took place annually on the 15th of Ramadan, the full moon on the middle day of the holiest month in the Islamic Calendar. The previous day the Sultan would take part in the ritual cleaning of the chamber and in preparing the mantle for display. Contemporary ‘Books of Ceremonies’ give precise details of the ritual - who was to participate, with their order of precedence sketched in diagrammatic fashion, what they must wear, and the prayers to be performed. [ref: Osman's Dream - Caroline Finkel, source: Necipoglu, Architecture, Ceremonial and Power]

Besides this specific yearly ceremony, the cleaning of the chamber which held the Prophet’s (S) belongings was a routine ritual of the Ottoman Sultans. The Ottoman Sultans are described as getting on their hands and knees and sweeping the floor and belongings.

They would then ask to be buried with that dust which had touched the Prophet’s (S) belongings.