SYNCHRO Construction Solution - After rescheduling, the task dates don't match those in my source schedule (P6, Asta, Microsoft Proj

Check your settings in Navigator>Project>Details:

Check the Data Date - should usually be set to Manual with the last schedule update as the date
Check the setting for Schedule - should usually be set to ASAP
Check the Schedule From option - should usually be set to Data Date

You may additionally check your settings in Options>Gantt Chart>Rescheduling>Parameters. Select the Help button in the Options dialog to learn more about each of these settings.