In this answer I will look at the geographical movement of the political entity known as Saxony. The history of Saxony can be divided into 5 periods Old Saxony (also known as pre-ducal saxony), Duchy of Saxony, Electorate of Saxony, Kingdom of Saxony, and Free State of Saxony.

During the time of Old Saxony, unlike all other periods Saxony was not a political entity with clearly defined borders. Since at the beginning of the first millennium Saxons didn't have a written language all written sources about the origin of the Saxons are written by Romans. The first mention of Saxons in a primary source is Ptolemy writing about the Saxones tribe during the 2nd century. According to Ptolemy the Saxones lived near the northern parts of the river Elbe.

During the following centuries the Saxon tribes migrated slightly Westward until they inhabited the lands directly south of modern day Denmark. When the Saxons were conquered by Charlemagne near the end of the 8th century their lands are described as bordering the rivers of Ems, Eider and Elbe.

After their conquest by Charlemagne the Duchy of Saxony was created in 804 AD. The Duchy of Saxony was ruled by the Liudolfing, Billung, and Supplinburg dynasties from 804 to 1137. During this period the borders of Saxony didn't change all that much. In 1137 the Welf dynasty which also ruled over Bavaria(in south-eastern Germany) became the Dukes of Saxony. They ruled over Saxony until 1180. When emperor Frederick revoked Bavaria and Saxony from the Welfs. Frederick seperated the Archbishopric of Bremen and the Duchy of Brunswick-Lüneburg from Saxony, and gave the remainder of Saxony(the southern parts) to the House of Ascania.

In 1260 after the death of Albert I, his two sons(Albert II and John I) jointly ruled the duchy of Saxony. Eventually in 1296 the duchy was split between the two brothers with Albert II ruling over the Duchy of Saxe-Wittenberg, and the sons of John I ruling over the Duchy of Saxe-Lauenburg. Lauenburg was in the north of germany on the coast. Saxe-Lauenburg split up into even smaller duchies and then reunited, and eventually was annexed by Prussia.

Saxe-Wittenberg on the other hand was centred around the town of Wittenberg. Which was on the southern most parts of Saxon territory. Parts of this territory was ruled by the House of Ascania even before they became Dukes of Saxony. Which they used to rule over with their title as the Count of Anhalt.

Due to the Golden bull of 1356 the dukes of Saxe-Wittenberg became one of the 7 people with the authority to elect the Holy Roman Emperor, and Saxe-Wittenberg became known as the electorate of Saxony.

The borders of the Electorate of Saxony moved further south in 1423 when it merged with the Margravate of Meissen and Thuringia. When the Margrave of Meissen Frederick I became the Elector of Saxony after the death of the previous elector Albert III without any children.

The 1635 Peace of Prague gave Lusatia to Saxony moving the borders of Saxony eastwards.

The electorate of Saxony also broke up into various different territories, but eventually those territories united, but I won't elaborate on that here due to the fact that it's not too relevant to the movement of the borders of Saxony.

During the dissolution of the holy roman empire the electorate of Saxony became the Kingdom of Saxony in 1806. Due to Saxony supporting Napoleon during the war of the 6th coalition the congress of Vienna in 1815 gave the northwestern 40% of Saxony's territory to Prussia.

The remainder of the Kingdom of Saxony eventually became a part of the German Empire in 1871, eventually the German empire became the Weimar republic in 1918, and the Kingdom of Saxony became the free state of Saxony.

Sources:

Saxon Identities by Robert Flierman

Vienna 1814 by David King

Renaissance and Reformation by William Roscoe Estep

As a follow-up, what about Niedersachsen and Sachsen-Anhalt?