I'm curious as to when the Egyptians stopped wrapping their dead in thin strips of fabric and transitioned to burial shrouds. Any historical speculation as to the reasons for this transition would be appreciated as well.
TL;DR: Roughly around the 3rd century AD, because Christianity.
Here is an article that briefly describes the historical development of ancient Egyptian mummification practices, including their decline.
In the Late Period and Ptolemaic Period (525-30 B.C.), the technical proficiency of the embalmers began to decline. During the Roman occupation of Egypt (30 B.C.-4th century AD.), the attention of the embalmers was focused almost entirely on the outer wrappings of the mummy. These elaborate wrappings, however, masked the evermore careless treatment of the remains. With the rise of Christianity in Egypt, the art of mummification, along with the ancient religious beliefs which governed it, gradually declined.
A Wikipedia article about Roman Egypt further highlights the influence of Christianity:
During the 5th and 6th centuries the Eastern Roman Empire, today known as the Byzantine Empire, gradually transformed itself into a thoroughly Christian state whose culture differed significantly from its pagan past... The triumph of Christianity led to a virtual abandonment of pharaonic traditions: with the disappearance of the Egyptian priests and priestesses who officiated at the temples, no-one could read the hieroglyphs of Pharaonic Egypt, and its temples were converted to churches or abandoned to the desert.
Mummies in the Roman period are known for their realistic facial portraits. According to a Wikipedia article about these portraits, "they were mounted into the bands of cloth that were used to wrap the bodies," and their production "reduced considerably since the beginning of the 3rd century." However, this article also emphasizes that relatively little is known about the final decline of earlier burial practices and that there were complex reasons involved.