I would have thought American liberals (like me) would mostly view race as a social construct and therefore not immutable. However I am not an expert in sociology and my experience must be shaped by not being part of any visible minority.
For example, take the example I gave above. People will not accept self-identify of "race" and have their own ideas of what someone "really is". In America it is often based on the one drop laws of segregation: someone whose ancestry is predominately white can be considered "black".
Also, consider the term I mentioned, coined by liberal Americans. It embeds the racist assumption that the world is divided into white and non-white people: that everyone who is not white one group.
A lot of liberal Americans argue against this.