They were called “bustle cages”, as they got…

.. heavier, stiffer, and began to form the shape of the dress.  This was called the “first bustle era” which would come and go pretty quickly between the mid 1870’s until about 1879 when they would disappear in favor of lightweight “tournures” during the natural form era, and then reappear as the huge “horse rear end” 3rd bustle era.