How do bodhisattvas, and those who would emulate them, respond to tyrants?

Since you're asking Buddhists, taking our point of view into account is important. According to the Buddha, we've all suffered brutal, grisly fates beyond count.

'Long have you thus experienced stress, experienced pain, experienced loss, swelling the cemeteries — enough to become disenchanted with all fabricated things, enough to become dispassionate, enough to be released."' https://www.accesstoinsight.org/tipitaka/sn/sn15/sn15.003.than.html

The Buddha also describes the amount of time we've been suffering these horrendous fates.

'“Suppose there was a huge stone mountain, a league long, a league wide, and a league high, with no cracks or holes, one solid mass. And as each century passed someone would stroke it with a fine cloth from Kāsī. By this means the huge stone mountain would be worn away before the eon comes to an end. That’s how long an eon is. And we’ve transmigrated through many such eons, many hundreds, many thousands, many hundreds of thousands."'

https://suttacentral.net/sn15.5/en/sujato?layout=plain&reference=none&notes=asterisk&highlight=false&script=latin

What I take away from that is that no matter how horrendous an experience we or someone we know have/has in this life, similar things have been happening to us for a truly indescribable amount of time.

Another aspect of this is the Buddha's simile about being dismembered and how you should feel toward your attackers.

'"Monks, even if bandits were to savagely sever you, limb by limb, with a double-handled saw, even then, whoever of you harbors ill will at heart would not be upholding my Teaching."'

https://www.accesstoinsight.org/tipitaka/mn/mn.021x.budd.html

The main thing to take away from this in my opinion is that Buddhist thought is a reaction to experiencing suffering over a duration that's not something we can truly comprehend. When we hear about someone being murdered horribly, part of the offensiveness of that is the idea that they had their "one chance" at life taken away horribly. The Buddha's word does not agree with that and describes a world where we've all experienced that countless times.

The Buddha prescribes loving kindness to LITERALLY everyone, no exceptions. That sounds like saying you should accept people doing horrible things to each other, but you can oppose them while still having empathy for any aggressors. You can oppose people non violently and still not harbor hatred or animosity toward them.

I'm a huge proponent of Tonglen practice, which captures the heart essence of Bodhisattva action.

When we witness the actions of despots, violence, and global atrocities— this is a call to practice.

“We must surrender our hopes and expectations, as well as our fears, and march directly into disappointment, work with disappointment, go into it, and make it our way of life, which is a very hard thing to do. Disappointment is a good sign of basic intelligence. It cannot be compared to anything else: it is so sharp, precise, obvious, and direct. If we can open, then we suddenly begin to see that our expectations are irrelevant compared with the reality of the situations we are facing.” ― Chogyam Trungpa

How I have been personally practicing through the war in Ukraine, and the January 6 Insurrection, is described in this piece: https://sunyata.info/writing/2022/3/25/a-prayer-for-ukraine.

Monks, who are professional practitioners of the Dhamma are expected to be completely pacifist.

"Well then, Punna. Now that I have instructed you with a brief instruction, in which country are you going to live?"

"Lord, there is a country called Sunaparanta. I am going to live there."

"Punna, the Sunaparanta people are fierce. They are rough. If they insult and ridicule you, what will you think?"

"If they insult and ridicule me, I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't hit me with their hands.' That is what I will think, O Blessed One. That is what I will think, O One Well-gone."

"But if they hit you with their hands, what will you think?"

"...I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't hit me with a clod.'..."

"But if they hit you with a clod...?"

"...I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't hit me with a stick.'..."

"But if they hit you with a stick...?"

"...I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't hit me with a knife.'..."

"But if they hit you with a knife...?"

"...I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't take my life with a sharp knife.'..."

"But if they take your life with a sharp knife...?"

"If they take my life with a sharp knife, I will think, 'There are disciples of the Blessed One who — horrified, humiliated, and disgusted by the body and by life — have sought for an assassin, but here I have met my assassin without searching for him.' 1 That is what I will think, O Blessed One. That is what I will think, O One Well-gone."

"Good, Punna, very good. Possessing such calm and self-control you are fit to dwell among the Sunaparantans. Now it is time to do as you see fit."

Then Ven. Punna, delighting and rejoicing in the Blessed One's words, rising from his seat, bowed down to the Blessed One and left, keeping him on his right side. Setting his dwelling in order and taking his robe and bowl, he set out for the Sunaparanta country and, after wandering stage by stage, he arrived there. There he lived. During that Rains retreat he established 500 male and 500 female lay followers in the practice, while he realized the three knowledges and then attained total (final) Unbinding.

Then a large number of monks went to the Blessed One and on arrival, having bowed down to him, sat to one side. As they were sitting there, they said to him, "Lord, the clansman named Punna, whom the Blessed One instructed with a brief instruction, has died. What is his destination? What is his future state?"

"Monks, the clansman Punna was wise. He practiced the Dhamma in accordance with the Dhamma and did not pester me with issues related to the Dhamma. The clansman Punna is totally unbound."
SN 35.88

The Buddha when faced with the bandit and murderer Angulimala, employed psychic powers apparently, in addition to his wisdom in the method of teaching.

Carrying his robes & bowl, he went along the road to where Angulimala was staying. Cowherds, shepherds, & farmers saw him going along the road to where Angulimala was staying, and on seeing him said to him, "Don't go along that road, contemplative, for on that road is Angulimala: brutal, bloody-handed, devoted to killing & slaying, showing no mercy to living beings. He has turned villages into non-villages, towns into non-towns, settled countryside into unsettled countryside. Having repeatedly killed human beings, he wears a garland made of fingers. Groups of ten, twenty, thirty, & forty men have gone along that road, and even they have fallen into Angulimala's hands." When this was said, the Blessed One kept going in silence. ....

Then Angulimala saw the Blessed One coming from afar and on seeing him, this thought occurred to him: "Isn't it amazing! Isn't it astounding! Groups of ten, twenty, thirty, & forty men have gone along this road, and even they have fallen into my hands, and yet now this contemplative comes attacking, as it were, alone and without a companion. Why don't I kill him?" So Angulimala, taking up his sword & shield, buckling on his bow & quiver, followed right behind the Blessed One.

Then the Blessed One willed a feat of psychic power such that Angulimala, though running with all his might, could not catch up with the Blessed One walking at normal pace. Then the thought occurred to Angulimala: "Isn't it amazing! Isn't it astounding! In the past I've chased & seized even a swift-running elephant, a swift-running horse, a swift-running chariot, a swift-running deer. But now, even though I'm running with all my might, I can't catch up with this contemplative walking at normal pace." So he stopped and called out to the Blessed One, "Stop, contemplative! Stop!"

"I have stopped, Angulimala. You stop."

Then the thought occurred to Angulimala, "These Sakyan contemplatives are speakers of the truth, asserters of the truths, and yet this contemplative, even while walking, says, 'I have stopped, Angulimala. You stop.' Why don't I question him?"
MN 86

While violence and destruction is strongly discouraged in Buddhism, it is ok for a bonafide ruler or government ("wheel-turning monarch") to establish police and armed forces to:

• protect and guard the people
• ensure peace
• ensure that justice prevails

This generally applies to lay persons and not monks.

‘But sire, what are the noble duties of a wheel-turning monarch?’

‘Well then, my dear, relying only on principle—honoring, respecting, and venerating principle, having principle as your flag, banner, and authority — provide just protection and security for your court, troops, aristocrats, vassals, brahmins and householders, people of town and country, ascetics and brahmins, beasts and birds. Do not let injustice prevail in the realm. Pay money to the penniless in the realm.
DN 26