If the task start date is not recalculated, it could be either because of Task Mode or constraints.
Do the following checks and changes
A. If you want Project to calculate the dates always. 1. Double click the successor. Set the Task Mode to Auto Scheduled in General tab. 2. Double click the successor. Set the constraint type in Advanced tab to “As Soon As Possible”. This is only for auto-scheduled tasks.
B. If you want Project not to calculate the dates but you want to adjust based on your preference 1. Double click the successor. Set the Task Mode to Manually Scheduled in General tab 2. Before you set the link click File, Options. In Schedule tab, check “Update Manually Scheduled tasks when editing links” to recalculate the date only once. 3. If you have already set links to Manually scheduled successor task and want to dates to be adjusted, select the successor task and click “Respect Links” in Task ribbon.