"The characters have to be true to themselves. The events need to follow a logic that fits the story. A single flaw and the magic is gone. The story dies." ~ Sam Lake
When a writer crafts their work they know that the story will not stand on it's own. More often than not, their work will be a piece of a series. While each piece needs to be complete in-of-itself with the ability to stand on its own, it's little more than a chapter of the larger story. Just like the singular story, the series will be it's own ride with its' own pacing, its' own suspense, tension and excitement. Unfortunately the most difficult part of crafting a larger story from the pieces is sustaining the continuity, the connections, between the pieces.
In the past, when a writer published a series the time delay between one piece and the next could exceed a year or more. This delay meant that when the next piece was released all of the recurring elements including setting, characters, themes and such would need to be re-introduced to the same detail that they were in the first piece. In addition to re-introducing the elements from the first in the series, the changes and events of the previous story(s) must also be reflected. Now, the delay between pieces has been shortened sometimes down to a matter of months between each. This means fewer details are re-introduced and repeated. Yet, there are still jarring transitions between most pieces in a series.
The largest failing for most when crafting a series of works is maintaining the same elements that are observed in the introductory piece. The tone can shift from one work to the next or more jarringly, the characters shift dramatically. While they act within the framework of the current story and it helps propel the current story, they are completely different characters from one piece to the next. These elements shift so dramatically that they do not match the overall work.
While some works were never intended to be part of a larger story, most were devised as continuing 'adventures,' a means of taking the favored characters and carrying them on so that the audience has more time with them. The strongest series are treated the same as other artistic mediums. Diptychs, Triptychs and so forth. One piece feeds into the next and while they are strong alone, they are even stronger when viewed as part of that series.