Transferring websites is something we do every week (almost every day actually) here is some example website transfer information we sent to a client:
Your domain is registered and hosted with us, there is three parts to it:
Name Server, A Records and MX Records, you have chosen to change your A Records (IP Address) but you have our Name Server and MX Records so hosting fees are applicable.
We were last contacted by the website design company, on your behalf, to change your A records on the 6th June and done that instantly for you.
Happy for you to transfer away or stay with us, if you transfer away there is a small transfer charge to pay (other charges ) and if you stay with us also happy. I can transfer the domain to another host but need their IPS tag and the £25 fee to complete this. In terms of existing emails, if you use an email client for your email then your emails will be stored for you locally and you will retain your emails.
We look at sites as if they were our own and can say that your site has been neglected in terms of SEO and can see some very amateurish SEO mistakes on it, which if you want to take our SEO program we will be glad to fix. Additionally we have now got more SEO successful projects that we can report on to give you confidence that we can deliver SEO success in terms of visitor numbers and search engine rankings. The first part of our SEO project involved benchmarking and reporting on the current status so we can use that as a reference for you and to show the improvements made over the coming weeks and months.