Is the story of Imam Malik kissing the grave of the Messenger of Allah, may Allah bless him and grant him peace, authentic? And if so, what is its source?

We have not come across anything that proves the attribution of this story to Imam Malik. What is known of his methodology is his avoidance of innovation and his blocking of the means to forbidden acts. The Imam can be asked about the reference for this story to verify its authenticity, and we do not think it originated from Malik.