Character Evolution Throughout a Series
To start, I think we first need a distinct impression of our character at the beginning of his journey. What does he want? What’s preventing him from getting that? What flaws are we working with here? Is there anything he can overcome?
Once we figure that out, we can work in gradual change throughout the series. A great way to look at it is, how do we grow as people? We experience a variety of situations and encounter a variety of people. The same proves true for our fictional counterparts.
Example 1: My heroine started the series with a penchant for meddling and letting curiosity get her in trouble. Over the course of the series, she really felt the consequences of those tendencies. It helped her mature as a person and opened her up to new opportunities... and relationships.
Example 2: My hero started the series with major trust issues. It wasn’t until my heroine blasted into his life like a hurricane, turning everything he thought he knew upside down, that he developed an interest strong enough to outweigh those issues. It gave him a reason to reevaluate—it gave him a reason to change.
If you ask me, things should escalate a little more in each book, forcing your characters to face to new challenges and trials. Then you can sit back, smile, and say, "The end." :)