The Revolutionary War certainly did create refugees. Manhattan and Long Island were occupied from 1776 to 1783 by the British, who took over a lot of the buildings, including residential mansions for use by the military. All the known Patriots had to flee, or else end up in military prison for the duration of the war.

Many of the New York refugees relocated north up the Hudson, as was the case with future Supreme Court chief justice John Jay and his parents. Many of the Jews of New York left for Philadelphia, only returning after the war had ended.

Similarly, during the Siege of Boston in 1775-76, a lot of Bostonians lost their homes, and were displaced to the surrounding area, and weren't able to return until the siege was ended and they could begin to rebuild.

Further, since Congress was broke, they paid their Continental Army soldiers in part by confiscating land from known high-profile Loyalists and giving it to Patriot soldiers. Many of these Loyalists moved in with family members, while a large number of them resettled in Canada at the end of the war, where the British government offered them free land in payment for their war services.

But there was little infrastructure set up in most of the Canadian land that the UK government offered these refugees, and a lot of them almost starved to death. 1784 was called "The Hungry Year" in Canada because there was a drought that summer and many people died over the winter. The experience inspired poetry and other rememberances.

The Land Ordinance of 1785 created allotments for Canadians refugees loyal to the US. This area was called the Refugee Tract. Approximately 70 households claimed land within the area.

Fun fact: the southern boundary of this tract is still known as Refugee Road, in Columbus, Ohio.

Specifically, in the Missouri-Arkansas area during the Civil War, saw a movement of refugees out of those states due to extremely violent guerrilla activities and military orders.

Missourians, both pro-Union and pro-Confederate, moved out across the state, but most from the "Little Dixie" area in northern Missouri, along the Missouri River. The refugees typically moved to other parts of the state, but would also move to other states like Iowa and Illinois for pro-union refugees and Arkansas, Texas and Oklahoma for the pro-Confederates. Also in Missouri, General Order 11 called for the forced evacuation of 4 counties along the Kansas-Missouri border, to help quell the violence, but also to help the Union army catch or kill pro-Confederate guerrillas.

Arkansas also saw movement of refugees, but not quite as much as Missouri. Most of the refugees were from northern Arkansas and moved to the same places for the same reasons as the Missourians. However, there were some pro-Union refugees that moved into southern Arkansas, but eventually returned to the area as Union cavalrymen to help capture or kill the guerrillas that drove them away.