In fifth-edition Dungeons & Dragons, “monstrosities” are monsters that don’t fit neatly into another category: they’re not beings from other planes, they’re not magically animated objects, they’re not animals or ambulatory plants, they’re not humanoids, they’re not giants or dragons, and they’re not undead. Mostly, they’re magically created beings, or they were at some point far back in time. In some respects, they behave like evolved creatures (the more so the longer they’ve been around), but they may retain oddities of behavior that reflect their unnatural origins.
The first monstrosity I’ll examine is another D&D classic, the owlbear: a creature of grizzly bear–like size and shape (and temperament) with an owl-like head. Applying my usual Str-Dex-Con analysis to it places it in the category of “brute”—high Constitution, extremely high Strength, relatively low (though still above-average) Dexterity—which means its preferred fighting style will be direct assault. Owlbears are dumb (Intelligence 3) and won’t prioritize one target over another. They’re not stealthy, but they’re also hard to fool with stealth, having Perception +3 and advantage on Wisdom (Perception) checks based on sight or smell. They’re pretty fast as well, able to outrun most player characters. Finally, they have standard darkvision, so we can assume that they’re active primarily at night and/or underground. An owlbear encountered in its lair during daytime hours will be sleeping, or at least resting, though the presence of other creatures may wake it up.
Owlbear tactics are going to have no sophistication. An owlbear that senses prey will Dash after it if it’s more than 80 feet away (the noise of which will prevent it from surprising the PCs). Between 40 and 80 feet away, it will move 40 feet toward its target, then Ready
a Multiattack an Attack action against the first creature that comes within its reach. At less than 40 feet away, it charges in and Multiattacks. Each Multiattack consists of a claw attack followed by a beak attack.
The owlbear will single-mindedly attack the first target it sensed unless and until it’s struck by a melee attack, in which case it will switch its focus to the last PC who’s done melee damage to it. (It doesn’t understand ranged weaponry.) A trained owlbear—such things apparently exist—can be commanded to keep attacking a specific target, but I’d apply a check similar to that for maintaining concentration on a sustained spell: a Wisdom check with a DC of 10 or half the damage inflicted by a different melee opponent, whichever is greater. On a failure, it reverts to instinct and attacks the PC who just wounded it rather than the one it was commanded to attack. (On a natural 1, it turns against its handler.)
In open wilderness, an owlbear will flee when it’s seriously wounded (reduced to 23 hp or fewer). If it’s cornered in an enclosed space, however, it will keep fighting to the death.
If an owlbear in the wilderness reduces one of its opponents to 0 hp and no one hits it with a melee attack before the start of its next turn, it will seize the unconscious PC in its beak and Dash (action) back to its lair with him or her. It won’t do this in an enclosed space or if it’s in its lair already.