How did colonial and British interests both dovetail and diverge in the motives of the French and Indian War.

The motives of the colonies and the British dovetailed because
they both wanted to defeat France. From there, the interests of the mother country and colonies
diverged.

The British were interested in defeating France worldwide.
The French and Indian War was merely one small part of the overall Seven Years' War. The British
were going to do whatever was necessary to secure their overall empire rather than doing things
that would be explicitly meant to advance the colonies' interests. For example, one reason the
British wanted the French out of America was the fact that the French were trading too much with
the colonies, bypassing Britain and defeating the mercantilist point of having
colonies.

By contrast, the colonies simply wanted France out so that
the colonists could expand into areas controlled by France. The British did not really care about
this. In fact, after the war, they issued the Proclamation of 1763 to prevent
this.

The colonies wanted the war to be waged for their own benefit
while the British wanted to wage the war as part of an overall plan to defeat France. Because of
this, their motives diverged a great deal.