Immigration is a good example. Cultural relativism has become broadly accepted among the progressive left. As a result, they increasingly can only see immigration as an economic issue. Preserving the country’s existing culture to them isn’t a valid reason, so they deem it “racist.” This is internally consistent to them. If culture doesn’t make a difference, the only reason not to have mass immigration is potential economic downsides, or limited infrastructure. Assimilation and the “Melting Pot” is increasingly coming under attack for the same reason.
This is totally incomprehensible to conservatives, who reject cultural relativism. They believe the success of a society is based on deep-seated cultural features and social reinforcement thereof. (I once had a Japanese person go through hundreds of years of Japanese history to explain to me why the country has less problems with corruption than many other Asian countries.) Within that framework, “we don’t want our community to change” is a completely valid position. At the very least, assimilation is indispensable.
And no wonder the right feels persecuted, because their beliefs are not necessarily hateful yet they are labelled as such by default. It's like drawing the line somewhere arbitrary and then asking the side whose beliefs are on the other side "why can't you discuss this without resorting to hate speech", and "stop complaining because your side is hateful" and "it's okay for companies to refuse you service because your ideas are hate speech and X-ist".
Few examples of acceptable "hate-speech" that falls within that acceptable line: Anti-white Racism. ACAB. Punch-A-Nazi.