Is there any source for the idea that if you speak Lashon Hara about someone then you get all of his sins and he gets all of your Mitzvos?
Yes. This idea is first recorded in the Chovos Halevavos of Rabbeinu Bechayeh, and from there it is recorded in many Sefarim of Gedolei HaPoskim and Mekubalim, including in the writings of the angelic teacher of the Beis Yosef. The original idea brought there is as follows: “One of the pious men stated: Many people will come to the day of judgment and when they are shown their accounts of merits will find many merits that they did not perform I will ask as to how it got into their list of merits. They will reply to them that in truth someone else did these merits and because they spoke badly of him therefore he merited to receive them. Likewise, when they notice that some of their merits will be missing they will be told that this is because they spoke bad against another. Likewise, some will find in their list of sins and transgressions that they never performed and when they will ask as to how it got onto their list they will be told that it was added to them because they spoke badly against the individual who transgressed them.”
Sources: See Chovos Halevavos Shaar Hakenia 7; Orchos Tzadikim Shaar Beis Anava; Maharit in Tzafnas Paneiach in name of Mabit; Maggid Meisharim Vayakhel; Reishis Chochma Anava 5; Shelah Shaar Haosiyos Reish; Chida in many of his Sefarim, see Alfa Beisa Kamaiysa; Chasam Sofer Derush for Zayin Adar; Rameh brought in Devash Lefi; Lachmei Todah; Shemiras Halashon of Chofetz Chaim 1 Shaar Hazechira 7; Michtav Eliyahu p. 214; Many Sefarim in Alfa Beisa Kadmaisa p. 81