Complete the last mission in the Point Lookout main story arc, "A Meeting of the Minds". Note that several missions take place between this and the "Walking With Spirits" mission. In this mission, you arrive back at the Calvert Mansion to see it go up in flames. You report back to Desmond in his built-in bomb shelter, and he asks you to follow him to the Lighthouse, where Professor Calvert's brain is taking refuge. Follow him the Lighthouse, but be sure to stick close as he doesn't move unless you're near. The Lighthouse laboratory is guarded mostly by Turrets and other various robots. Fight through the lab, into the room where Calvert's brain is. You can either listen to his babble and go through with killing Desmond for him, or you can silence him yourself; your choice.
Noted frustration: After battles, Desmond will not move unless you first progress a little. Don't wait around for him after battles.