In Persuasion, do individual characters manage to move out of the classes they were born into? If so, how?
The short answer to your question is - yes, individuals can change class. Sir Waler Elliot and Lady Russell are old-fashioned snobs who don't think you can change class, but their views are challenged in the novel.
Admiral Croft and Captain Wentworth have both amassed fortunes which allows them to move into another class - though they are treated with suspicion by the likes of Siir Walter.
English society was changing at the time the novel is set. The wars against Napoleon had just finished. The fact that Croft and Wentworth are both naval officers is crucial. I will explain.
At the time to become an officer in the ARMY took no special skill and rich young men could buy their way into the army - the army thus reinforced the class system. The navy was very different. Before you were put in charge of a ship, you had to demonstrate leadership qualities: ships were expensive and men from poor backgrounds could be promoted in the navy because of their skills and merits - unlike the army.
What has this got to do with your question? Well, in the navy if you captured an enemy ship, you made money - the implication is that Croft and Wentowrht have done well in the wars and made enough money (through their indivual skill as sailors) to move into another class.
Austen allows Anne to marry Wentworth and presents the attitudes of Sir Walter and Lady Russell as ridiculous and old-fashioned. Hope this helps!!