There's nothing wrong with a culture adapting over time without losing their identity, like your example of Brits deciding to add the kebab. But a small country can't preserve their identity when they are overloaded by so many immigrants coming in at once who FORCE their ways on their new home. Think of it as Push vs Pull, there's a difference.
Every country having a unique culture makes the world more interesting. Imagine how boring it would be if you saved up for a trip to Japan/Kenya/Chile/etc, only to find that almost everyone there was a white English-speaking American living the exact same lifestyle you have at home? Leftists would in that case be empathetic toward the remaining indigenous minority who feel their historic way of life was killed off against their will. Why should it be any different if you swap the country and nationalities in this example to what is happening in the UK?
I am an immigrant myself, and a fairly liberal one at that. But I made the effort to come to my new home legally and assimilate to the best of my ability. I'm not sure why I should hold others to a lower standard.