Best Practices

One of my clients is terminating his Comcast/Xfinity service and switching to Verizon. He has 5 years of emails (about 5.0 GB of data) and is worried about losing them. We attempted to transfer the emails to a Google account, but only a fraction of them were successfully moved. He’s considering not switching to Verizon because of this issue. How can I ensure that all his Comcast emails and photos are transferred to his Gmail account? I am not a technical person and need a step-by-step guide for this process. Thank you!