Yes. In 1941 a group of Iraqi fascist sympathisers called the golden square attempted a coup against king Faisal II who was backed by Britain along with his regent and government. At this time Britain had held Iraqi as a protectorate under the mandate system about a decade prior and it still maintained influence there with military bases and stake in the Iraqi oil industry. Pretty soon after the coup fighting broke out around the RAF bases in Iraq.

Among the men who guarded the bases the Iraqi Levies. These were troops drawn from Iraqi ethnic minority groups, predominantly Assyrians but a little under a 1/4 were Kurds to, who were part of the RAF with the primary job of guarding RAF airfields in Iraq. The Iraqi Levees took part in the defence of the RAF Habbaniya and RAF Shaibah until British relief forces arrived to help defeat the Golden Square's coup forces.

After the end of the Anglo-Iraq war many of these men continued their previous duties, guarding airfields though some of them were transferred to Palestine and Cyprus to do it. They did however also form a paratrooper company some of which were kurds who went on to serve as more frontline troops.

I've been able to find out little about any kurds who might have fought on either side in the allied invasion of Vichy controlled Syria.