As to the specific question, I don't know the definitive answer, but, as a student pilot, would you really want to be taught by someone who "spreads their favours" over several airfields, is only partially familier with the full working practices of the flying organisation you are paying £000s to learn to fly with and who, whilst "teaching" you, may well be desperate to get to the next airfield where more lucrative renumeration awaits.
Most instructors are "mobile" in that they will move on to other organisations, but I've not come across any who are "freelance" in the true sense of the word.