To take a more natural example, האיש הולך למסעדה is a bit different from האיש הולך אל המסעדה - the latter emphasizes the walking, while the former glosses over the mode of getting there (in English it would be "go" rather than "walk").
Now, האיש רץ אל המסעדה vs. האיש רץ למסעדה. It's harder to see a difference, since רץ is rather specific about the mode of getting there; but still רץ למסעדה could be used figuratively, if the man read about the great new restaurant and he anxiously hurries there.
Now replacing "man" with "dog"... I can no longer imagine a difference.