The pick-offs at baron were bad, but Gambit was already trying to come back from a lost game at that point. Gambit lost the game because of their lane swap vs a Cait-Nunu lane. Cait-Nunu can be played as a poke lane or a trading lane depending on the matchup, and it can preform well in both ways. But there is one thing that Cait-Nunu excels at, and that is quickly pushing towers. With Caitlyn's range and Nunu's bloodboil, they can hit tower for a huge amount of damage at a safe distance, and Caitlyn's netshot provides her with a valuable escape. In a 2 vs 1 situation, Caitlyn can safely attack only from the side bushes to gain stacks of her passive more quickly, and her Q also provides excellent waveclear.
This pushing power isn't so much of an issue in a 2v2 lane, and the reason is that the support can help counterpush the lane; Lulu and Sona are good choices against Cait-Nunu in part because they are both fairly good at pushing, and both of them can punish Nunu if he tries to help Caitlyn push the lane too much. But there's no way Miss Fortune and Lulu are going to push tower nearly as quickly as Nunu/Cait can, and so by laneswaping their ADC/support to mid, Gambit basically gave up a free turret (and then didn't freeze lane to continue shutting out Ryze).
Here is a link to the beginning of the game in the VOD.