I don't know about 5 years but at BA they only recruit Dispatchers internally and as it is possibly one of the very few companies who still seem to respect the role. it is still a prestigious job, which means very few people leave etc.
They have recruited dispatchers recently and quite a lot have reasonable service although i know that some have less than 2 years service, but previous experience.
You would most certainly have to work on Check in or the ramp first and the starting rates of pay in these areas is not that great anymore although you do get the airline perks if you can afford to use them !
Many of the handling agents also recruit from within but they do accept from outside as the standard of internal candidates is not good enough due to the salaries on offer etc.
I am referring to Heathrow, what happens elsewhere i wouldn't know about, but having worked at Heathrow for 16 years in various roles including Dispatch / Load Control / ops etc, i wouldn't think any other airport would provide the Challenges of LHR.