When this word arrived in Middle English, it soon developed a range of senses, including the modern one, and we find this in Shakespeare. At the very end of Macbeth (V.vi.105), Malcolm expresses his intention to call home "our exiled friends abroad" (ie from outside Scotland). But in most of Shakespeare's uses it has no such connotation. When Falstaff says to the Lord Chief Justice, "I am glad to see your lordship abroad" (Henry IV Part 2, I.ii.94), he means only "out of the house"; they are not in some foreign country. And even more general senses are found. When Derby says to Audley in Edward III (II.ii.21) "the king is now abroad", he means "on the move".
And when Curan says to Edmund, "you have heard of the news abroad" (King Lear, II.i.7), he simply means "everywhere".
David Crystal is author, with Ben Crystal, of Shakespeare's Words, published by Penguin