Summary of the long answer to “Are there any particles whose invariant mass ever changes?”: The decay width thing is very real and could mean a “yes” to your question, but the center of the invariant mass distribution is fixed and thus could mean a “no” to your question. The quark mass stuff points to a need for more context when talking about individual quark invariant masses at all; that is, the first question isn’t “is the invariant mass the same” but rather “what even is mass here?”