Catch the Legendary Dogs
Use your Pokedex to track the legendary dogs. When you go to the entry for the legendary dog you are trying to catch, you should see a button labeled "Area." When you push this button, it will highlight the route on your map that the legendary dog is on. Because flying to a city will cause the legendary dog to move to another random spot, do not fly to its location. A better strategy is to pick one city and repeatedly fly to it. Check your Pokedex each time, and if the legendary dog is near your location, chase after it on your bike or on foot.
Status Conditions
Put the legendary dogs to sleep or paralyze them immediately. Put a Pokemon that has a move such as sleep powder or grasswhistle first in your inventory. Also, make sure the Pokemon has extremely high speed. A Sceptile that has the egg move grasswhistle is a very good pick. Putting the legendary dog to sleep will also prevent it from fleeing from battle, allowing you to reduce his hp more than one move at a time. Stun spore is the best paralyzing move. Sleep is preferable to paralysis, but paralysis will also allow you to fight the legendary dog longer without it running. Always inflict a status condition before you try to catch any of the legendary dogs. Other status conditions such as burns, poison and confusion are not recommended. They damage your target and may cause the legendary dog to faint, rendering it uncatchable.
Strategies
Carry a pokemon that knows false swipe. When the legendary dog is at low hp, use false swipe to reduce its hp to 1 without causing it to faint. The lower the hp, the higher your chance of catching it. If you do not wish to waste your Master Ball, use Ultra Balls and Timer Balls. Other Poke Balls do not provide the catch ratio to capture the legendary dog in a reasonable amount of time.