Hello Helga,
The only designated unit of West Kents serving in the Third Afghan War of 1919 was indeed the 1/4th Battalion, which was comprised mostly of territorial soldiers. However, men from other battalions, who were in India at the time, en route from various theatres of war awaiting repatriation, were sent up to Quetta, to reinforce them. One of them was my own grandfather, who had been in Mesopotamia, serving with the 2nd Battalion. Perhaps your great-grandfather was similarly drafted in to reinforce the 1/4th Battalion.
The 1/4th Btn saw active service in Afghanistan, and distinguished themselves in the Battle of Spin Baldak, which resulted in the capture of the massive, and heavily fortified strategic Afghan stronghold.
I have much more information, including some photos, which might be of interest, so send me a pm if you'd like more details.
Regards,
Swampy
