The simple answer is that migration levels were probably near historic lows during the wartime years of 1776-1783. Part of the problem answering a question like this is that the migration statistics that people think of in the modern era didn’t exist in the period c. 1760-1820. People think of annual totals of immigrants listed on registers or censuses, but these were rare. Philadelphia has a few series of registered servants and Redemptioners in the Revolutionary period, but they are spotty at best during the war years (Eg. Record of Indentures of Individuals Bound Out..1771-1773. Baltimore, 1973 or Strassburger, Pennsylvania German Pioneers). Farley Grubb has done extensive work measuring European emigration to the US that spans across the revolutionary war years, and most of his work is forced to exclude the war years themselves due to poor data (Grubb, “Servant Auction Records and Immigration into the Delaware Valley, 1745-1831” Proceedings of the American Philosophical Society, 1989). Part of the reason for this drop in emigrants is practical, as you mention. There was a full scale war going on in the colonies, and therefore travelling there was unappealing. However, in the years leading up to the war, emigration was among the highest it had ever been, even after the “opening of hostilities” at Lexington and Concord in New England in 1775 (Bailyn, Voyagers to the West).

A larger factor in the why emigration was lower during the war years was because British ships were largely prohibited from trading (and thus carrying passengers) to the colonies. The fledging colonial government had put an embargo on British goods, and in response, Britain forbade trade with the rebellious colonists. This killed most incentives to send ships to the colonies, as it was both dangerous and economically speaking, potentially disastrous. The vast majority of emigrants that sailed to the North American colonies from 1760 to 1775 came from the British Ilses. Perhaps as many as 125,000 migrants sailed from Britain and Ireland to North America between 1760 and 1775 (Bailyn, pp. 25-26). While this estimate is doubtless too high, it does help reflect the dominance of the UK in the years leading up to the war. About 12,000 German-speaking emigrants disembarked in the Delaware River valley (mainly Philadelphia) in the same period, with perhaps 3,000 more arriving in other ports (Wokeck, Trade in Strangers and “The Flow and the Composition of German Immigration to Philadelphia, 1727-1775” Pennsylvania Magazine of Hist. & Bio. 1981). The greatest number of known migrants during the war years were actually English and Hessian soldiers who decided to remain in the colonies after the war ended – perhaps as many as 3,000 Hessians chose to stay in the US after the fighting had ended (Fogleman, “From Slaves, Convicts, and Servants to Free Passengers” Journal of American History 1998, Appendix A).

Perhaps more significantly, though less considered, was the fate of the other major group of Trans-Atlantic migrants; enslaved Africans. The British slave trade mainly operated between Britain, Africa and the West Indies, but a significant number of slaves were either shipped directly to North America (typically to the Chesapeake or South Carolina) or purchased in the West Indies and shipped to the northern colonies via a slave factor (O’Malley, Final Passages and "Beyond the Middle Passage” WMQ 3rd Series, 2009). In 1774, the Continental Congress passed a resolution agreeing to prohibit the importation of slaves after December 1, 1774, and to a large degree this ban was effective (Journals of the Continental Congress, Volume 1, p. 77). Besides the ban on slave importation in North America, the American and French Navies caused pandemonium in the British Atlantic. Slave ships were always high risk-high reward ventures, and the war years made the slave trade that much riskier. American of French navy vessels could capture and overwhelm the crews of British slavers. There is also a lot to be said about the credit/risk system of British payments for slaves, but this is not the space for an exceedingly technical discussion. The simple reality is that the British slave trade went from disembarking 37,000 captives in 1775 to 9,000 in 1778. Risk of capture, high rates of credit, and non-importation agreements all factored into what the famed abolitionist Thomas Clarkson called an “experiment with respect to the abolition of it [the slave trade]” during the War for American Independence (Slave trade data taken from the Trans-Atlantic Slave Trade Database online at slavevoyages.org. Clarkston, History of the Rise, Progress, and Accomplishment of the Abolition of the African Slave-Trade, 2 volumes, (1808) 2:374-375).

So, to sum up, emigration to the North American colonies reached historic lows during the years of the War for American Independence. There was likely a larger outmigration of people than in-migration; Loyalists sailed to Britain, the West Indies and Canada in droves. The British slave trade all but ceased, and imports into North America were halted altogether. The greatest number of emigrants into the colonies were most likely British and Hessian soldiers arriving to fight the war, instead of settle and multiply. However, as Aaron Fogleman has emphasized, the character of migration to North America shifted from 1775 to 1783. What began as a market of dominated by unfree convicts, slaves and servants came to be replaced by emigrants travelling without the burdens of indentured servitude, and the convict trade to North America was replaced with New South Wales by 1787-1789. The slave trade would continue until its final outlawing in 1808, it became a less tasteful trade, and only really had a resurgence in South Carolina from 1803 to 1807. The emigration to the newly formed United States began to take on the ideals we more commonly associate with it; that of opportunity and a land of liberty.