If all of your tables' records had timestamps, you could identify "the values that have changed in the server" -- otherwise, it's not clear how you plan to do that part (which has nothing to do with insert or update, it's a question of "selecting things right").
Once you have all the important values, somecursor.executemany will let you apply them all as a batch. Depending on your indexing it may be faster to put them into a non-indexed auxiliary temporary table, then insert/update from all of that table into the real one (before dropping the aux/temp one), the latter of course being a single somecursor.execute.
You can reduce wall-clock time for the whole job by using one (or a few) threads to do the selects and put the results onto a Queue.Queue, and a few worker threads to apply results plucked from the queue into the internal/local server. (Best balance of reading vs writing threads is best obtained by trying a few and measuring -- writing per se is slower than reading, but your bandwidth to your local server may be higher than to the other one, so it's difficult to predict).
However, all of this is moot unless you do have a strategy to identify "the values that have changed in the server", so it's not necessarily very useful to enter into more discussion about details "downstream" from that identification.