BA didn't ever fly to Karachi during the 767's near three decades in the fleet - that was a service that stopped in the early 1980s if my memory serves me right, and Karachi used to be a TriStar over Dubai.
Post the VC10 era into Rawalpindi, BA had the 747s on LGW-ISB and then LGW-MAN-ISB BA2119 routing for a few years. It then went to LHR on 747 and latterly 777 before being dropped ten years ago. Unless I'm much mistaken, there was a flight ops technical reason why BA 767-300s couldn't do Islamabad - something about the RR-powered aircraft dry operating weights being heavier and a consequent problem with drift-down altitudes for those specific aircraft being below the MSA over one of the mountain ranges sticks in my mind as the reason why. One of the BA folk did explain it to me once upon a time during a discussion about BA 767s based at Manchester and why they never went east. It was certainly a convoluted problem but one which didn't affect routings going further south for the likes of HYD and BLR which - although even further - did receive BA 767 service over the years.
Happy to stand corrected if anyone can be more definitive!