In every beat-em-up there are moments when the tactical situation turns bad. So the designers give you the equivalent of a shmup bomb to get out of trouble.
I'm not talking about the emergency spin-kick that costs you 15% of your life. That's an entirely local effect for when you're really screwed. I'm talking about the larger picture, affecting a majority of the screen.
One method is the "magic spell" method, where you are given a tool with a limited number of shots. Whether it's a fireball or SWAT intervention, these serve two goals simultaneously. 1) They deal damage outside your normal range, and often higher than you normally can. 2) They knock down enemies at range and give you a moment to maneuver.
An example of a spell-heavy game would be the old Dungeons and Dragons beat-em-ups (Tower of Mystara, for example).
Another method is "throwing". Throwing locks your target while you act, and serves the same two goals as magic spells. It deals damage to the character you are acting against, and then you can usually hurl them to a point outside your normal range. In the process, they knock down all the other enemies they are pushed into.
Throwing serves the same role as a magic spell, but the restrictions are higher (requires nearby enemy, slower) and you can do it whenever. Also, I feel that using one enemy against another is fundamentally more interesting than just tossing magic death everywhere.
The obvious example of a throw-heavy game is Streets of Rage.
A rare option is the "dodge". This option basically makes you invulnerable for a short time and maybe moves you somewhere else on the screen. In some ways, this is the purest method, as it doesn't magically solve your tactical problem. It just lets you try again once the wave has passed.
Dodging is extremely rare, probably due to the chaotic nature of beat-em-up screens and the fast wind-up of the attacks. The only two that spring to mind are God Hand and Astro Boy.
Most beat-em-ups implement one of those methods well, and one shittily.
For example, the Dungeons and Dragons beat-em-ups did spells pretty well, but they also implemented blocking, which is a type of dodge. However, the blocking is shitty.
As another example, Streets of Rage implemented dodging really well. However, they also implemented a 'spell' - the SWAT team backup. Unfortunately, the SWAT team ended up being mostly pointless: the only use it has is to use against the bosses to knock off a bar and a half of their radically over-inflated health bars. Using it during the level is rarely a good option.
Plenty of bad beat-em-ups do two of them, both poorly. This may or may not be what makes them bad, but the correlation is high. Golden Axe implemented both magic and grappling, but implemented both in a very limited fashion. Whether you think it is a good game or not, you will admit it is a rough play.
So, how can you avoid implementing them poorly?
To implement spells correctly, variation is good, but not critical. The most critical thing to remember is that it shouldn't have a dominant use.
For example, in Streets of Rage, the only real use for SWAT backup is to soften up bosses. They are so useful for that purpose that wasting them on killing off dudes is considered really crappy form.
While the bosses are also good targets for spells in the D&D beat-em-ups, many spells are only useful in other situations, which spreads the pain around. To be honest, I feel that even these games are too boss-centric. These old games built their bosses to eat quarters rather than be interesting challenges, so obviously your resources will be at least partly reserved for fighting the sure-to-kill bosses.
To do throwing well, you need to have very few enemies (if any) that are actually immune to throwing. In most Streets of Rage games, literally every enemy can be thrown, including bosses. If you make immune enemies, you are sealing away a core move. It's the equivalent of an enemy that magically makes you drop your assault rifle in a first person shooter.
This is somewhat difficult if you want to do a fantasy beat-em-up, because it means the player would have to be able to grab small enemies and large enemies, ground enemies and flying enemies... so fantasy beat-em-ups usually play down grappling.
It should be noted that there's a difference in opinion as to how throws should affect multiple enemies. In classic Streets of Rage, if you grabbed an enemy in a pack of enemies, your grappling attacks would affect all the enemies. The actual throw might not, but the pre-throw head-smashing would.
However, in the remake, you will let go of whoever you're grappling after a single pre-throw attack, and I don't think you can throw them at all when you're grappling in a crowd. This has seriously weakened throwing, and moved my old favorite character (a throw-heavy character) to the bottom tier.
Personally, I feel this is a bad move, because I tend to marshal the enemies into a pack and then crush them with throwing. So, it totally blocks me.
As to dodges, I'm not sure about them. It's rare to see examples so there's not a lot to draw from. It seems like dodge-based beat-em-ups are very different. They're less about a complex tactical topology, and more about timing.
There, a technical post about beat-em-ups.