Scotland never really stretched to the Humber. The Kingdom of Northumbria at one time stretched from the Humber to the Forth. The southern part of Northumbria (Deira) became part of the Danelaw whilst the northern part (Bernicia) remained outside as a client state. Much weakened the northern half of Bernicia (from Forth to Tweed) was taken over by the emerging Scottish kingdom. In the east of the country apart from Berwick itself the border was pretty much aettled by the Battle of Carham in 1018. The Scots lay claim to the rest of Northumbria and the early wars resulted mostly from Scottish aggression and English retaliation but it was never officially part of the kingdom. The Scottish Kings did officially control Cumbria in the west for a briefish period. At a slightly later date the Scottish monarchs held large swathes of land in what is now England but that land wasn't part of the Scottish kingdom. For that land they swore fealty to the King of England.