Web hosting is a very organic business. Clients come and go all the time. Businesses come and go. Good ideas survive and less successful ones need to be abandoned. Sometimes a move to a new web host is urged on by your web designer. Maybe your designer is more comfortable with her regular web host or, as is quite often the case, your designer is getting a kickback for referring new business. You are now being told to move your site in order to put a few extra bucks in your web designer's pocket. Whatever the motivation, many designers will also tell you that you need to transfer your domain name when you make the move.
Whatever decision you make, you should be aware that moving your hosting business to another provider is more easily done when you do not transfer your domain name. Domain name transfers can be complicated, can involve a lot of frustration and are often just not worth it. You may say that I have a bias here, because I offer both web hosting and domain name registration. For the record, let me just say that I like to make it clear to new clients that they don't have to transfer their domain registration to wundersolutions.com when they move the web site here.
If you choose not to transfer the name, all that is involved is you logging in to your domain management tool and changing the nameservers. This is typically two text boxes which you have to fill in. Click the submit button and you're done. A transfer, on the other hand, involves a purchase of at least one additional year for your domain. It may require filling in an extensive form with the new domain name reseller, requiring an authentication code from your current registrar, waiting for a confirmation email and going to another page to confirm the transfer. Then, once the transfer is complete, you still have to log in to change the nameservers.
It's a complicated and convoluted process. You don't do it often enough to remember the steps. So, next time you move, you'll probably not remember how you did it the first time and the fun begins anew.
You should also keep in mind that many registrars make little to no money on domain name registrations. They're typically "loss leaders", or low priced items designed to attract your initial business so that you'll spend money on hosting, SSL certificates, email etc. My advice? If you're happy with your current registrar, don't bother transferring your domain name. If you're unhappy, by all means, find a new registrar. Just know that it's not a requirement, regardless of what you're told.