I am convinced the developers of thus games stat and level sysyem are super autsists the likes the world has never and may never see again.
Everything in this game affects a stat.
The items, the losition of character mext to each other while fighting, their levels, the monsters they fight and their levels, the attacks you use and are hit with, the environment this is taking place in as well as they order they are killed and the mission its on.
Say you fight with a sword but also know a heal spell with light magic. You do that enough times and you become the games version of a paladin
BR gain seems to be affected by various factors. Number of enemies in the encounter (and number of links/Calvary Calls you have to suffer through), maybe how long it lasts, how many units/unions fall during battle and how many units are left standing. I think all of those are taken into account for how many "points" are gained to push the BR up. The question now is, at what point does BR go up? Does the gap get wider per BR gained? Does BR difference, which is used for multipliers, affect how quickly it goes up? Is each set of encounters a set number of "points" that is then modified? Most of these points can be ignored if you're not going to poke at the finer details.
As for stats, you can influence them based on the units' actions. Be careful about this since it can also influence their focus through whatever commands you give the union. All require the unit to be alive at the end of battle to receive the upgrade.
HP: Get hit. Don't be afraid to get them a little roughed up.
AP: Use arts. Well, more like spam them.
STR: Use physical attacks (normal, CAs)
INT: Use MAs
SPD: Use IAs/healing, make them run laps around the battlefield, scoring a CT (a CT popping up and missing won't count)
UA: Use IAs/healing.
A note about the CTs… Those are affected by the equipment and current army morale.