A method copyeditors use to edit nonfiction is to read from end to beginning to catch mistakes they might otherwise miss. The technique can work for fiction, too.
For those literalists out there, this isn't meant to advocate reading a work backward word by word, but to start with the last chapter or paragraph and read the prior. Here's what to look for and why.
Timelines that don't track: Characters' ages must work within the time frame of the story and should mesh with the ages of other characters. Events should occur in sequence, like children being born after their parents. Even when you intentionally present events in an order other than chronological, it's important that they occur in some logical order. Actions, too, should occur in logical sequence. You'd be surprised how often characters leave a building before opening the door, and how often they repeat an action within a sentence or two of a scene.
Changes in tone, mood and voice: Whenever time elapses between when a writer begins and finishes a work, changes can occur in tone, mood and voice that don't fit the evolution of the story, and indicate a shift in the writer. Common problems are humor occurring where it shouldn't because it reflects the writer's frame of mind, and characters becoming a lot less sophisticated or more so, depending on what the writer did with the plot. A change in authorial voice is perhaps the biggest problem, and often occurs when the writer grows more accomplished to the point where it shows in the piece.
Character development/arc: One of the most important aspects of a story is the development of each major character, either evolving or devolving, depending on the storyline. Regardless of which occurs, it's important to see this happening and feel it. An uneven arc can give the work an uneven quality, as when a character suddenly goes from good to bad or bad to worse with little development to support the change.
Reading a story from end to beginning can reveal these problems, especially to the writer who's on the lookout for them.
Tip: Read a short piece or a scene from a story you're working on, and list the inconsistencies. Then list possible fixes and where these must occur in the work.