The first two emails bridge the conversation from LinkedIn, provide a bit more information, and provide a way to get on the phone. The second email is your chance to go into more detail – both on your understanding of your prospect’s business, as well as how you can help them.
The 3rd email introduces a sense of urgency which can often spur your prospect into action.
The 4th, especially when done with a bit of humor, can demonstrate that you are determined to help your prospect while also providing a chance to showcase your personality a little more.