While the direct effects of the Norman Invasion, such as the change in king and transformation into a Norman French culture, remained confined to England, the event left a lasting mark on late medieval Scotland. The main example of the enduring influence of the Norman Invasion can be found at Malcolm III's court, as in 1068, he sheltered English (Anglo-Saxon) refugees. In 1070, the Scottish king married a member of this exiled group of English nobility: Margaret, granddaughter of Edmund Ironside. More importantly, their youngest son David I would return to the English court and reform Scotland in turn.
Through Malcolm's marriage to Margaret, the Scottish court began to experience a shift. Primary sources (such as the Vita Margarita) comment how the new queen began to 'civilise' the barbarian Scottish king, which in historical reality meant that she began to reform the Scottish Church in order to conform to practices elsewhere in Europe. While Anglo-Saxon England was therefore being converted into a bastion of Norman French society, Scotland was opened up to influenced from the continent through Margaret. After Malcolm's death, three decades of changing kingship ensued, but David I, his youngest son, ascended the throne in 1124 and continued to reign until 1153. From his childhood onwards, he had been part of the future English king Henry I's household, and he had been taught in the Anglo-French styles of ruling. For instance, he held lands in Northumbria and was even granted the earldom of Huntingdon through marriage. Like his mother, he came to Scotland and reformed: but where his mother had only transformed the Church, David I also changed the political systems and the economy of Scotland. He welcomed immigrants, both merchants and knights from France and Flanders, settling them in advantageous positions. It will surprise nobody that these transformations stemmed from ideas that were circulated at Henry I’s court.
To conclude: there were no immediate effects of the Norman invasion, but the immense changes in England found their way to Scotland by way of Queen Margaret and King David I.