The running Pokémon are the most irritating captures in the game. You can't simply save before encountering them. You have to get lucky enough to find them, then manage to capture them!
Entei, Raikou, and Suicune can be found roaming the Johto region (Gold/Silver/Crystal). Latias or Latios can be found roaming in Hoenn (Ruby/Sapphire/Emerald). Either Entei, Raikou, or Suicune can be found in the Kanto region (FireRed/LeafGreen). The requirements and recommendations are below. You must be lucky to run into each one or trade from another game.
Use the Pokédex to find their current locations. Fly to a central location with either three or four routes leading outward. Check your Pokédex. If the runner is not within two routes of you, go to the border of the city. Every time you cross that line, the rare Pokémon moves. When you cross it enough to be in the city and to be within two routes of you, check its distance. If it is two routes away, move in the direction towards it. If you are one route away, use another exit from the city (into another route without the rare Pokémon). The Pokémon may come into the same route or move away. If it moves away, start all over. But, you might be in the same route. Note: If you save and continue later, the rare Pokémon will move.
Once you manage to be in the same route as the given runner, use a repel with a level 40 or under Pokémon in the first position of your party. As long as the area you're in doesn't have Pokémon at or above level 40, you'll run into the rare Pokémon. Use a Master Ball if you have one. Why? None of the other rare Pokémon run away (like Lugia and Rayquaza). If you don't use it, you've got to do it the hard way. First of all, they don't heal when they run away, so don't fear that. You may want to use Mean Look (or some other battle-sustaining move) to keep the battle going or use Sleep Powder (or another sleep-inducing move). After prevented from escaping (for a little while), cut down its HP. If it wakes up, the battle might be over. When down to a sliver of health, use a Poké Ball (Fast Balls recommended).
Mesprit and Cresselia can be found running around as long as you've met them at least once (Diamond/Pearl/Platinum). In Platinum, the three birds can also be found running around after you've been referred by Professor Oak: Articuno, Zapdos, and Moltres. Mesprit and Cresselia can be found at level 50. Articuno, Zapdos, and Moltres are found at level 60. Instead of using the Pokédex, use the Marking Map app on the Pokétch. The current locations of all runners will be shown there. Use a similar strategy as Generations 2 & 3 to locate them. This time, however, inducing Sleep will NOT keep them in the battle! Throw Quick Balls when their health gets low enough.
In HeartGold and SoulSilver, these Pokémon can be found as runners: Entei, Raikou, Latias (HeartGold only), and Latios (SoulSilver only). Instead of using the Pokétch, however, you'll be using the PokéGear. Like before, every time you switch between map screens, they'll move. Check the Marking Map on the PokéGear and see where they've gone. Use a similar strategy as before.