Balancing this game entirely will be basically impossible on the grounds of availability unless you make growth rates basically nonexistent so you have leave behind your earlier units in favor of new ones. Assuming you want to allow us to keep basically one team throughout the whole thing, I'll come up with some quick ideas. Some of them will probably suck. Deal with it. I'm coming up with them on the spot.
Try making Franz and Seth the cavalier duo and nerf them from there as necessary, maybe.
Eirika's probably pretty good.
Try making Gilliam a soldier with a fair but more SPD and then give it some promotion options, maybe?
Vanessa and Moulder are probably good as is, considering that it's hard to nerf their utility.
Maybe nerf Garcia a tiny bit and then make Ross a level 1 Pirate/Fighter
Give Neimi kickass offense. This may need to change somewhat in light of other earlygame units getting nerfed.
Either buff Colm's combat or make it harder to replace him via keys.
Artur and Lute are probably good. A slight nerf might not be a terrible idea for Artur. Lute needs either a buff or a nerf depending on playstyle.
Natasha's in the same boat as Moulder.
Either hurt Joshua's earlygame or his lategame. He's not great right now, but with so many earlygame units getting nerfed, he could become pretty awesome if he isn't touched.
Ephraim could use a pretty substantial nerf. Maybe take Ephraim and his knights all back to level 1 (with bases to match) and then weaken 5x enemies to match them?
Bump Amelia up to at least a level 1 cavalier. You may need to go farther from there.
Tana's probably in a pretty good spot. This may change after you make changes to others, though.
Looking at Gerik's group, I'm not entirely sure what to do with Innes. He might be good. As for Gerik and Marisa... maybe drop Gerik's level some and then bump Marisa up to level 10? I don't feel particularly good about those solutions, but at a glance even a 10/1 Marisa would only have as much STR as a level 2-4 Gerik, although she would have +1 MOV, +15 CRIT, and like +5 SPD compared to him.
Dozla's probably pretty good. Hard to say where he'd be after you edit other units.
Give L'arachel at least the opportunity to insta-promote.
I'm not entirely sure what to do about Saleh... I'd look at decreasing weapon levels (especially staves) as well as lowering durability or maybe screwing over his growths so he isn't very viable by endgame. I like my prepromo sages, so I'm not to keen to think of ways to make them not awesome.
Ewan should probably be brought up to at least nearing promotion to tier 2. His 10/1/0 stats are still pretty bad for that point in time.
Cormag could just use an all around nerf. Lower everything some.
Rennac's probably in a similar position to Colm.
Nerf Duessel. Nerf almost as much as Seth.
Knoll... bump up instantaneously to Summoner perhaps with a good staff rank?
Myrrh's probably decent as is.
Syrene would probably like a slight buff in the combat area, but the general nerfing of characters in this game might make her a solid combatant as she is now.
Again, some of these ideas are probably terrible. Just throwing them out there.