Wow, when I watch the NCAA basketball games I see that many times the teams who are already winning do not slow down and just continue to score many points. They are adding insult to injury. In other words, the losing team is already losing the game, and they come in piling up more points with no regrets. Listen here.
Generally referring to police officers or firemen or military who lose their lives in the line of duty, their work, this idiom can also refer to you. Listen here.
Someone who is always moving and always into something new is like a stone that moves. There is an old saying that stones that are moved do not get moss on them. So like the stone a person who is always moving is not considered set in their ways.