In many occupations this is forbidden by contractual terms and carries with it a stiff penalty.
Not depending on how it is phrased when you talk to the customer -
Allowable - I am leaving XYZ company later this month and look forward to keeping in touch with you when I settle in to my new role.
Typically not allowable - I am leaving XYZ company to go to ABC company and want you to shift your business over there.
Add in another layer about any NDA/Non-compete he may have signed. Depending on the severity of the non-compete, a sales role typically isn't allowed to get to his new job and just go straight to his old client list and switch them over. HOWEVER, and this is why LinkedIn is beautiful... connect with all your customers on LInkedIn before you leave, let them know you are leaving, then when you switch to the new company and update your LI profile --> will show up in the feed of your old customers. They see it there, reconnect with you.
You haven't specifically targeted them and asked for their business. They've come to you.