What happened to Donalbain and Fleance after the war between Macbeth and Macduff?
Malcolm, the eldest son of Duncan, becomes King. Donalbain is the second son, so he would be next in line to the throne should something happen to Malcolm. However, mostly, the second son is expected to go into the church as a clergy (which is actually what was planned for King Henry VIII of England before his older brother Arthur died) or into the army. Something that would bring honor to the family, but would not detract from the importance of the first-born son serving as King to the nation.
Fleance, the son of a noble, would also follow in the footsteps of his father, Banquo. He would be one of the chief go-to men for the King--warrior, advisor, helpmate, friend, confidant. In return for his loyalty, he would expect to be paid by titles, lands, wealth, and good marriage for himself and his children. As a member or nobility, he can also be ambitious enough to seek the throne for his children through marriage or appointment as was foretold by the witches.
However, this is all speculation. We don't know exactly what happened to them as the play ends with Malcolm's "thank you" speech to his loyal followers which brings the play full circle since it opened with a battle and Duncan's "thank you" speech to his loyal followers.
In 2.3 upon learning that his father has just been murdered, Donalbain tells his older brother, Malcolm, that he fears their lives, too, are in danger. Malcolm decides to flee to England, while Donalbain decides to flee to Ireland. It is assumed that Donalbain remains in Ireland, for he is not mentioned again. Malcolm assumes primary importance as he is the heir to the throne. Macduff is immediately suspicious of Macbeth and shortly thereafter travels to England to join Malcolm. In 3.1 Macbeth hires murderers to kill Banquo and his son Fleance. Although Banquo is murdered, Fleance escapes. The witches' prophecy that the royal family would continue through the line of Banquo is therefore confirmed as possible. Fleance is not mentioned again in the play.
I am not sure what happens to Donalbain but it is thought in some historical circles that James I of England, whom this play was written for, is a decendant of Fleance so it is likely that he or his son became king after Malcolm. This ties with the witches prophercies as they show 8 kings with banquo at the top with a mirror and possible James at the end with glass.