"Although most persons with diplomatic immunity carry diplomatic passports, having a diplomatic passport is not the equivalent of having diplomatic immunity. A grant of diplomatic status, a privilege of which is diplomatic immunity, has to come from the government of the country in relation to which diplomatic status is claimed. Also, having a diplomatic passport does not mean visa-free travel. A holder of a diplomatic passport must obtain a non-diplomatic visa when traveling to a country where he is not currently nor is going to be accredited as a diplomat, if visas are required to nationals of his country."
More frequently, I hear of UK embassies not offering full diplomatic protection to their non-political staff in foreign countries. If you're sending someone from your Tourism Board to the British embassy in Thailand, you simply don't want the hassle of having to specifically revoke his diplomatic privilege when he shows up in Pattaya with a dead hooker, where you're happy to give them diplomatic car plates so the traffic police don't try and get 500THB from them for speeding... I assume this is an international trend, rather than something the Brits thought up.