Ground Support Equipment (GSE) from the major world Ground Handling Companies do move their GSE quite often. This is to either meet the requirements at a particular airport/for fleet replacement or for deep maintenance. This often occurs not only with in the UK but also between Ireland and other parts of Europe. (Brexit warning re import duties and taxes notwithstanding perhaps).
Generally there are two main pieces of equipment that are not easy to move around. Main Deck Hi-loaders and some types of towbarless push back tractors both due width. These normally are classed as wide loads and need careful planning and approval from police etc., before moving. Hi-loaders in particular are best not moved around as they are very sensitive to any kind of unusual movement. They can take some time in my experience in the past to settle down after a major road journey.