This process is commonly referred to as a Transfer of Ownership or TOO by way of an abbreviation.
Typically it is a fairly seamless process in which it doesn't require too many documents for you to fill in however this will depend on the network that bills you.
All major networks will require a credit check to be ran on the new company to establish whether they are eligible or not. Vodafone require a cover letter alongside a signed document stating that you are happy for us to process this. O2 only require the signed document alongside EE. Any Ice mobile customers are also only required to sign a document.