"Latest_Sync_Date" is a field in one of the client's tables, correct?
Then you can perform a find on the tables using this date as search criteria in a scripted find to pull up a found set of just the records you want to import into this specific client. You can set a variable to this value, enter find mode and then use set field with this expression: ">" & $Latest_Sync to set up a criterion based on the value of this field on the client.
You may want to see the scripted find examples found here for some ideas on how to script this: Scripted Find Examples
Yes, each table has a field called "Last_Update_Date". Any records with a date > than the oldest "Last_Update_Date" in my local tables needs to come over.
This just came to mind...should I be importing from the server file/tables themselves or should I add these source tables as table occurances inside my client file? Thus, when I find the target set of records to import, those are the only records to be referenced when I run my import script?
I think using external data source based table occurrences in your current file is the best option as it simplifies a number of details of your scripted process, but this method works whether or not you do that. The basic rule for Import Records is:
If the file is not open when Export Records is executed, you get all the records in the table. If the file is open, you get all the records for the specified table occurrence's found set. So as long as you perform the needed find first, (Perform script in File A, can pass criteria as a script parameter to a script in File B in order for the script in File B to find the records...), your import records step will import the found set rather than all records.