Moses took the bones of Joseph during the exodus because

Exodus 13:19 records: "Moses took the bones of Joseph with him because Joseph had made the Israelites swear an oath. He had said, 'God will surely come to your aid, and then you must carry my bones up with you from this place.'"

This oath dates back to the end of Joseph's life in Egypt. In Genesis 50:24-25, Joseph, knowing he was about to die, told his brothers that God would eventually bring the Israelites out of Egypt to the land He had promised to Abraham, Isaac, and Jacob. Joseph then made his brothers swear that when God visited them and led them out of Egypt, they would carry his bones with them to the Promised Land.

Moses honoured this oath during the Exodus. The reason was not that Joseph was a king (he was a vizier or prime minister, never a king), nor that it was a general Israelite custom to carry bones, nor that the bones had any intrinsic preciousness. The sole reason was the solemn oath that Joseph had extracted from the Israelites, binding future generations to transport his remains to the land of Canaan. Joseph's bones were eventually buried at Shechem (Joshua 24:32).