Wherein the home country in Europe is invaded, possibly by the French, and the royals have to go to the Colonies and continue from then on.
This AH cliché/overused trope is inspired by the exile of the Braganzas to Brazil between 1808 and 1822, after the Napoleonic invasion of Portugal in 1808. Predictably, these events had a massive impact on OTL 19th century Brazilian and Portuguese history, and often have a big or even bigger impacts in ATLs, if they occur.
A rather frequent variation on the general idea is that during an ATL WWII, the British royal family flees to Canada (as planned but not carried out in OTL, or that some anti-monarchical revolution occurs (republican-aristocratic, republic-democratic, or communist/syndicalist) and the surviving royal family has to flee to some royals-loyal part of the British Empire or British colony. Technically, the latter case did occur once in OTL history, when Oliver Cromwell banished the royals and made Britain into a quasi-republic/dictatorship, but the British royals fled to France and elsewhere in mainland Europe, rather than overseas, and the monarchy was eventually restored in the second half of the 1600s.
Other variations also exist. Rarer examples have even included the Russian imperial family being ousted from their homeland earlier than in OTL and seeking refuge in a still Russian-controlled Alaska, which they break away as a separatist Russian-speaking country, still loyal to the old Russian monarchy. There have also been French, Spanish and other monarchies in overseas exile.
As you can see, though it is somewhat of a cliché, different authors have often sucessfully put an interesting spin on the basic idea.