I originally planned to only use caught Pokemon throughout the story mode, but then I quickly realized that (at least when you're in Johto) the game doesn't like to give you very useful Pokemon to beat most of the gyms. I traded over a Makuhita and a Sneasel (with the evolution item) early on and they ended up being about 10-15 levels higher than anyone else because Fighting, Dark, and Ice-types are very useful in Johto. As I went through Kanto's gyms, my other Pokemon caught up in levels... except for my poor Feraligatr, who became redundant.
Only at first. Later on, it's no contest.
Falkner - Geodude
Bugsy - Geodude and Spearow, Pidgey as well.
Whitney - Trade a drowzee to a guy in the city for a Machop
Morty - Anything with a Psychic move, and definately anything with a dark move (pursuit, bite). You can get both TM28 (Dig) AND Pinsirs with Moldbreaker (no abilities work in battle against it) at the National Park.
Chuck - Psychics still ruling, and flying type attacks aren't that rare either. Magnemite can swap for Flaafy if Poliwrath is raping
Jasmine - Graveller handles the Mangemite easy, and Onix is even more destroyed by water since he's swapped Sunny Day for Sandstorm.
Pryce - Piloswine is a bit of a jerk, but the whole gym is made for giving you electric pokemon levels.
Clair - The only real "dick" move, since her pokemon are nearly level 40 and any ice type you caught prior is sitting at around 25. Lapras gets Ice Beam in the mid 30s, but for everyone else it's likely you're going to need to start Voltorb Flipping for enough points to get Ice Beam.