In A Tale of Two Cities, what one connecting link is there in which Darnay, the Manettes, and Mr. Lorry were involved?

I think you are referring to Book 3. In this chapter,
“Knitting Done” all of the dots are connected.  Dr. Manette was imprisoned because of
the baby he delivered for Charles Darnay’s uncle, the Marquis St. Evremonde.  He
condemned the Marquis and his whole family for the rape or impregnation of the young
woman.  This is the reason he has his relapse when Lucie marries Charles, because the
connection with the name and the events that sent him to prison is too painful.   As the
nephew, when his uncle is killed the title of Marquis is passed to Charles, making him,
his wife and his daughter nobility.  It is now a crime to be noble in France, and Dr.
Manette has signed his son-in-law’s death warrant with his letter condemning the St.
Evremondes, so Darnay is sentenced to death.  Mr. Lorry, as always, plays the role of
protector and facilitator of both the story and the family by getting Lucie, Charles and
the daughter out of France.