The cover stories relevance differs from character to character.
Buggy: Gets his body and crew back, teams up with Alvida, leads into Loguetown.
Coby & Helmeppo: The two get training from Garp, leads into the Post Enies Lobby arc with Garp revealing who he is.
Jango: Escapes Syrup Village, meets Fullbody, joins the Marines with him when they see Captain Hina. Leads into the end of Alabasta.
Hatchan: Escapes the Marines, meets Camie and Pappagu, leads into Sabaody
Wapol: Creates a toy empire, develops Wapometal which Franky uses for his post-timeskip tech.
Ace: Nothing really. It does show that Ace will get really defensive when people insult Whitebeard, which in turn leads to what kills him.
Gedatsu: Nothing yet. It does show Pell's body has been left heavily scarred after the bomb.
Miss Goldenweek: Nothing for her, but Crocodile, Daz Bones, Bentham and Galdino end up going to Impel Down, which leads to their presence in that arc.
Eneru: Nothing yet, but I imagine it will come into the story later.
CP9: Nothing yet, though if that shadow during the Punk Hazard auction was any indication, we might see them soon.
From the Decks of the World: Appearances from multiple minor characters post-timeskip, including proof of life of characters thought dead (Mr. 9, Miss Monday, Bentham).
I imagine that Caribou's story will lead into something, which will be paid off in the main story down the road. Especially now that he's on one of Kaido's islands, and the Straw Hats are after him.