Many white refugees ended up in Philadelphia and caused an out break of yellow fever. The end result was a major epidemic that killed 5000 of a population of 50000. Another 20000 would flee the city due to the outbreak. The revolt made many governments in the region on edge. The English, Spanish and the French fought the Haitian Revolutionaries. The loss of Haiti also caused Napoleon to sell Louisiana to the USA. Meanwhile commodity markets were thrown into chaos as Haiti was the largest producer of sugar in the world at the time. The Impact of the Haitian Revolution in the Atlantic World by David P. Geggus is a good book of you want to read more about it

Other colonial societies:

The Haitian Revolution was seminal in the outbreak of the Brigand War, which is sometimes conceived of as a spreading of the American Revolution or French Revolution into the Caribbean, especially the Lesser Antilles (the little islands closer to Venezuela and Brazil than to Florida and Louisiana).

The British and the French were engaged in hostilities during the period leading up to conflict, but then leaders like Victor Hugues ("the demon of republicanism") rallied armies of freedmen and ex-slaves against landowners, kicked out governors of colonial powers, and helped form a Caribbean identity....

On Saint Lucia, the brigands were known as "l'Armee Francaise dans les bois" - they established maroon communities in the rain forest.

Read more with Curtis Jacobs or Gerard Besson, who's written a little about Hugues, or St. Lucian historian Jolien Harmsen.

The years are a little strange here - in 1792, the slaves in Haiti were formally freed. In 1794, Hugues kicked the British out of Guadeloupe, and in 1795 out of St. Lucia... but they came back to St. Lucia in 1796. In France, Napoleon took power so that by 1799 or 1800, slavery was reinstated in Guadeloupe, and by 1803, slavery was reinstated in St. Lucia.

There was more fighting in Haiti in 1802, but the country defeated Napoleon's troops and declared itself independent in 1803.

So the Haitian Revolution wasn't strictly speaking a cause of the Brigand War or the other Caribbean conflicts in the period... but there was a relationship.