If a Minotaur character hits a creature with bludgeoning damage (say from a maul), it can use a bonus action to use its Hammering Horns feature.
From MMoM p. 27:
After you hit a creature with a melee attack as part of the Attack action, you can immediately use a bonus action to attempt to push a target within 5 ft' of you and no more than one size larger than you. It must succeed on a STR saving throw or be pushed up to 10 ft. away from you.
If the Minotaur character also has the crusher feat he can push the character 5’.
From Tasha’s p. 79:
Once per turn, when you hit a creature with an attack that deals bludgeoning damage, you can move it 5 feet to an unoccupied space, provided the target is no more than one size larger than you.
Because the crusher feat does not indicate the creature is moved “away”, it allows movement anywhere - even up.
From Jeremy Crawford:
Pushing someone away requires the whole move to be away from you. A diagonal push works. Vertical doesn't.
If the Minotaur was to bash the creature and it gets knocked 5’ straight up, the falling rule comes into play. Here is a clarification on the falling rule from Xanathar’s p. 77:
The rule for falling assumes that a creature immediately drops the entire distance when it falls.
So when the Minotaur hits its opponent with bludgeoning damage (maul attack) and uses the crusher feat to lift it 5’, both the bonus action Hammering Horns and the falling occur “immediately” after the attack (see above) and at the same time. But according to Xanathar’s, when 2 actions conflict, the player/dm controlling the character gets to choose what goes first.
From Xanathars p. 77:
Simultaneous Effects - Most effects in the game happen in succession, following an order set by the rules or the DM. In rare cases, effects can happen at the same time, especially at the start or end of a creature’s turn. If two or more things happen at the same time on a character or monster’s turn, the person at the game table — whether player or DM — who controls that creature decides the order in which those things happen. For example, if two effects occur at the end of a player character’s turn, the player decides which of the two effects happens first.
So the Minotaur could choose the Hammering Horns to occur before the fall. And since Jeremy Crawford has already said a diagonal push works, a 10’ diagonal push from the Hammering Horns feature would allow an opponent to be moved 5’ away and 8.6’ up.
From Pythagoras 570 BCE:
a=√(c2﹣b2) = √(102﹣52) ≈ 8.66025
So this brings me to my full question. Can the Minotaur attack using its maul, use crusher to knock it up 5’, then prioritize hammering horns and knock it 5’ over and another 8’6” up (now 13.6’ in air) and then let it fall (taking 1d6 falling damage and becoming prone)?