TL;DR - I think that what custom moves you get is linked to what character you're playing as AND what moves you use while playing that character.
So, I was looking through what custom moves I still needed to unlock, and noticed that I was missing at least 1 "counter" type move on almost everyone with a counter type move (Lucina, Little Mac, Shulk, etc). I almost never use counters in battle, and I wondered if there was a correlation there. So I conducted a test:
1) Lucina - Customs missing: Effortless Blade, Iai Counter
I went into All-Star mode on hard and just spammed the **** out of Counter for the first two stages. I ended up dying on round 2, but went back in and spammed it a bit less. I ended up getting Iai Counter after stage 3. Then, I did the same with Dancing Blade, and got Effortless Blade after stage 4 of my first run of spamming it.
2) Zelda - Customs missing: Farore's Squall, Din's Flare
This was my control group. I went in, and just played Zelda as I normally would (which is to say, not a lot of Farore's Wind or Din's Fire). I beat All-Star on hard, and got 2 Zelda customs that I already had. I did this again, and got 1 Zelda custom I already had. The third time, I spammed the hell out of Farore's Wind, and got Farore's Squall after stage 3. I was then able to get Din's Flare on a second run of hard All-Star after stage 2 by spamming Din's Fire.
It seems that if you want a specific custom move on a character, you can get it REALLY quickly by spamming the hell out of that "move slot" in All-Star mode on hard. This is great for people like me, who are missing 1 or 2 custom moves on a bunch of characters. I'm not sure if this works in other modes, but I'm pretty sure it does.