Hello Stuart and welcome to the forum.
I am assuming that the "City Line" in Liverpool is the one to Hunts Cross that continues on to Warrington Central and Manchester. If so, the "slamdoor" trains you refer to were probably Class 108 Diesel Multiple Units or even possibly Class 115's. If you follow the links from
this page you will find more details of both classes.
I could be wrong but I have a sneaky feeling that the mainline from Lime Street to Runcorn and Crewe was also called the "City Line" at one stage in which case the "slamdoor" trains could have been Electric Multiple Units of Class 304.
Picture here
Hope this helps you to track down some more info.