Not only is it possible but that's what the tool should be doing. The finish column is what will show the date change as that column is dynamic. But that assumes you did not hard code the finish date, i.e., put in a must finish on constraint. Your packages should be constraint free as much as possible such that the start and finish dates will constantly move to the left and to the right.
So if a predecessor is finishing late, then the start of the successor will start late accordingly and will show in the start column. You can also open a start variance and finish variance columns to see the number of days your schedule is shifting from baseline.
EDIT based on comment: Formal changes to the schedule occur on a predetermined frequency. In some cases this is weekly and in other cases it is monthly. This depends on the length of the project and how sensitive we need to be on tracking variances. There are ad hoc changes, too, but I attempt to minimize those. Therefore, tracking the changes becomes a product of passing time, e.g., we are 17 weeks into a 42 week project; therefore, the schedule was updated at least 17 times (assuming a weekly update).
Now when I say formal changes, I am not talking about a change that requires a re-baseline to the schedule. Changes here are only those where progress and actual dates or costs are loaded.
Changes that require a re-baseline are formally done through change control. The outcome of this process is formally documented with all the approvals, impacts, risk assessments, etc.
