Example. This claim seems to me to be drawing on two things. Firstly there's the alleged tolerance of Muhammad that turned to intolerance as Islam grew stronger in the early years (as Murray discusses slightly earlier in this video). Is this an accurate reflection of the accounts we have of the early years of Islam? Secondly, it's drawing on the comparison of Islamic political activism in the West (focusing on Islamophobia) with the more extreme political developments in the Muslim world which are notably intolerant.
However neither of these are really exactly analogous to what the anti-Muslims are warning against. Therefore I'm interested if there's any historical examples to back them up of this change occurring within one society. One thing I read a while ago by a blogger I respect says that this did happen, but he didn't provide any sources. He says:
"the author makes the case for the emergence of a "mélange" civilization. It would be Islamo-Christian, with antecedants in the early Ottomon polity. The problem I have is that such cultures have always existed within the Dar-al-Islam, but they tend to be anterior to the eventual development of a more thoroughly Muslim polity that reduces the non-Muslim population into dhimmitude. Of course, we are not slaves to history, Westen Islam could spark a Reformation in the faith, but I simply want to remind everyone that the rising Muslim elites have a long history of initial cooperation with the non-Muslim powers that be until the time when the dhimmi become disposable. Also, let me note that many of the dhimmi families that allied themselves with the early Muslim powers later converted to Islam and became part of the Muslim elite."
Perhaps he is referring to developments in the early years of the Ottoman empire?
I always find it ironic when people try to claim that Muslims are taking over a country via immigration and "breeding," and that this is apparently part of their religious beliefs no less. This is not the conclusion one would get if one actually looked at what orthodox Sunni Muslim jurists said about residing in non-Muslim polities. The permissibility of living under non-Muslim rule was historically a controversial issue. See, for example, Andrew March's discussion of this topic in Islam and Liberal Citizenship: The Search for an Overlapping Consensus:
Douglas Murray even said it himself in the video you've linked: "I'm not an academic." His personal interpretation of the Prophet Muhammad's career doesn't mean much considering that conservative jurists and modern Islamists such as Sayyid Qutb considered emigrating from non-Muslim lands as religiously mandatory. Aren't people such as Sayyid Qutb taken as true followers of Islam by people such as Douglas Murray?