If the main character does not change (as in some stand-alone series like Dracula or Geronimo Stilton), the other characters around him change and evolve. Some secondary characters can become one of the main characters in one of the sequels because they help advance the plot. In Twilight by Stefanie Meyer, Jacob barely appears in book 1, but takes a growing role in the sequels.
A sequel does not need to have the same character lead. In a saga, the lead character changes from book to book because the saga spans over multiple ages and eras. What links the stories is the world.
New characters can appear in the second or third book as long as the books are part of Act Two. Some characters can disappear for a while.
Write a story Bible where you include all the new terms, places, rules, maps, house plans, clothes, quirks or your characters. You need to keep track of their hobbies, tastes, the places they went to, the people they’ve met, in order to be consistent. Every detail counts because your reader will remember them. You might not remember the color of your heroine’s sash, but the reader will keep track of that.
Include shared objects or pets, talismans even, symbolic objects, and repeat some snippets of conversations to induce recognition and familiarity. It could also be images, metaphors or feelings people share across the timeline. In the Sisterhood of the Traveling Pants by Ann Brashares, a pair of jeans links all the characters of the series.
Make the enemy become the ally or at least the hero should seek the help of the villain to resolve a problem. You could also make an ally become the enemy or at least make sure an ally's behavior becomes questionable. Make sure the leaders change camps or put themselves in question in order to add drama and some dimension into the story.
If your hero has a handicap, keep in mind that it needs to be part of the plot and helps advance the story. It should not constitute a hindrance. Opting for a blind detective will limit his skills at observation and will make him dependent on someone else, but he might catch sounds other people cannot hear. Another problem is stereotypes if your character is for example an alcoholic? Too much emotional baggage might prevent the reader to identify with the protagonist and become redundant along the way. You embark in a long journey with the same heroes; make sure you can handle them.
An on-going love affair can spice up a series and be exciting if the romance is attacked, or impossible, or appears on and off.
Make your characters complex. The more complex, the more possibilities. In order to render a three-dimensional character, consider the character’s personality (multiple strengths and weaknesses, unique qualities, flaws or vices, showing that the MC is able to grow and be stretched thin); problems (a MC should have more than one problem to handle); goals (your MC need to change goals and motivations along the way, but still keep faithful to the strongest ones); hobbies.
Tim's Printables. "Examples of Satic VS Dynamic Characters."