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Well, it was brought up in the indictment section of the Declaration of Independence. So the colonists were clearly upset enough about it to consider it a good argument for their position.*
He has excited domestic insurrections amongst us ...
This doesn't explicitly say slavery, but that's what everyone understood they were talking about. Specifically about the very actions of Lord Dunmore mentioned in the question. The founders were (quite understandably) loathe to put the word "slavery" in print, so they tended to talk about it obliquely like this.
Dunmore's proclamation was issued after open rebellion in Virginia put him in desperate straits. On the order of 1-2 thousand slaves actually deserted to his forces, which certainly isn't an insignificant amount. By the first US Census in 1790 there were about 300,000 slaves in the entire state of Virginia, but that was likely a big increase from 15 years prior in Dunmore's era. So statistically it probably wasn't a big deal but they certainly bolstered Dunmore's forces. Worse, since slaves outnumbered white men in the state nearly 3 to 2, the implications if things progressed were quite scary (for the white men).
In the event, Dunmore did field actual regiments of escaped slaves against the forces of their former masters, but found he went from having to fight on the order of 150 "patriots" vs. his 100 regulars pre-proclamation, to nearer a thousand vs. his 450 post-proclamation. The former he could win with, the latter ... no. The rebel colonists drove Dunmore's forces to the sea, whereupon he was forced to abandon Virginia entirely, never to be allowed back.
So I think its fair to say this was in fact a big motivator in Virginia. Perhaps not so much in the lower South or the North where slavery wasn't so big (and the population ratios weren't so scary), but Virginia was the most populous colony (Pennsylvania a close second), so what happened there mattered a lot.
* - I've elided out the second part of this grievance, which complains about a similar effort to get Native Americans to fight the colonists. It used lurid language designed to appeal to the imaginations of Europeans, but which is frankly a disgusting embarrassment today. Since its highly insulting and off-topic, I won't repeat it here, but readers should be aware it is there.
Upvote:2
The Battle of Lexington was in April 1775.
The proclamation was in November, 1775.
There appears to be an impossibility here.
There was talk earlier, to be sure, as Burke mentions in his Conciliation, to discuss an issue with it:
must they not a little suspect the offer of freedom from that very nation which has sold them to their present masters?βfrom that nation, one of whose causes of quarrel with those masters is their refusal to deal any more in that inhuman traffic? An offer of freedom from England would come rather oddly, shipped to them in an African vessel which is refused an entry into the ports of Virginia or Carolina with a cargo of three hundred Angola negroes. It would be curious to see the Guinea captain attempting at the same instant to publish his proclamation of liberty, and to advertise his sale of slaves.
Parliament had, in fact, forced Georgia to allow the sale of slaves, so there were complications in making the offer.