He nods. 

Does anything happen on their way?

Not too much. They encounter some small songbirds singing at one point, and can hear various insects as well. A couple hours into the walk the sun has peeked up over the horizon and begins to shine through the trees, casting the inside of the woods into bars of light and dark. They'll pass a good foraging spot along the way as well, and Bran will ask if Emiya might like some berries or greens to replenish some energy expended on the walk so far.

"Yes," Kiritsugu says, grateful. "I will need you to point out the ones that will not poison me."

There are definitely some berries or herbs in the pines that are moderately to severely poisonous! Bran reassures that this patch of bushes and shoots is all good, though he'll also point out a couple plants that aren't good to eat large amounts of in one sitting. None of the berries are perfectly familiar to Kiritsugu, but the flavors of blackberry, blueberry, and black currant are all identifiable, and the primary herb that Bran gathers seems like a milder, sweeter, more tender leek. It's all a welcome reprieve from this long early morning trek. Soon, however, they are back to walking.

He's eaten a lot of food in a lot of different places so he is experiencing the taste equivalent of uncanny valley. 

This is nice. He's trying not to be suspicious of it. 

if Kiritsugu stays here for long, it will probably grow on him. Certainly it isn't bad in its own right.

It's another couple hours of hiking, the sun steadily rising, before the shape of the cave-sleeper's hill resolves through the trees. They are very close now.

He keeps walking without talking much, trying to conserve energy.

Bran likewise mostly stops talking.

The two men continue their approach of the cave, and only a few minutes pass before the mouth of the cave gapes before them. At the surface it does not seem especially large, but looking into its dark interior betrays a great uncanny depth. A gentle cool breeze emanates from within.

"I have the feeling we should not enter the cave itself," Kiritsugu says, feeling rather grim even just looking at the entrance. "I feel like we'd freeze to death, or worse. How are we luring the thing out?" 

He really wishes he had a rocket launcher. Rocket launchers make everything less complicated. 

Bran nods in agreement. "You should stand back, maybe find some cover for you to shoot at the sleeper from. I will get it's attention and try to keep it in the open."

He takes a couple steps to place himself right around the middle of the cave mouth, sets himself into a solid stance, begins to move through a series of breathing exercises. Kiritsugu can sense Bran's aura of magic begin to intensify.

Kiritsugu follows his advice and conceals himself behind low greenery. He can stand up quickly when necessary, but for now he's going to stay the hell out of the way. 

Bran might not be a heroic spirit, but he has their intensity, especially as his aura wakes up. He's reminded of Saber.

He is also less kindly reminded. 

Bran's magic continues to heighten until it begins to radiate visibly just beyond his clothes, a thin coat of golden glow that flickers and dances. Then, he inhales deeply, places his hands around his mouth with his thumbs and forefingers forming a diamond, and--

The light is almost blinding, the sound like thunder, a wave of hot air nearly bowling Kiritsugu over. Reinforcement quickly overcomes this and allows Kiritsugu to observe Bran's exhalation, a thin golden beam piercing the cave's darkness.

It only lasts briefly before Bran cuts it off and quickly changes his stance. In the next moment, the cave-sleeper, a seething mass of the same darkness the size of an elephant, extending unnaturally into the light of the outside world, charges into Bran. Battle is joined.

This is not what Kiritsugu usually thinks of when he hears "bear." It reminds him of the slick, angry darkness that came pouring out of the ruined Grail, corrupting everything. 

He takes out his gun with practiced readiness, and shoots the massive bulk of it in the side. He wishes he knew if there was a bundle of nerves, a head, anything. It's wielding its power. 

He watches the bullet travel. Hopefully it does what he needs it to.

The cave-sleeper can tell that a projectile of some kind has been fired. It does not try to dodge, it does not have any reason to believe that it is a threat. It is concerned with the loathsome fire-in-flesh which has so often been an obstacle between itself and the warm fire.

This is a mistake.

With a mighty heave, Bran wrestles the cave-sleeper such that the magic bullet strikes true and center. Jets of black liquid blast from the cave-sleeper's mass, evaporating into a claret mist and then into nothing as the beast roars and writhes in pain. The cave-sleeper attempts disengage with Bran but he holds to it tight. "That's more hurt than I've ever seen it!"

Looking closely, there are gaps in the cave-sleeper's cloak of darkness now, small but visible to Kiritsugu, that he can spy something more animal through, glimpses of fur and skin and blood.

If it is something like an animal it can die like an animal. A cloud of wasps, a Dead Apostle, a bear in a sea of solid darkness; all can burn. All can end. 

"I'm glad!" 

He fires again. 

The beast is certainly more wary of Kiritsugu's bullets now, having felt the sting of one already. It fights Bran with even greater strength, now that it knows it is in real danger. Bran is not able to force it in the way of the bullet again, and instead it also able to roll out of the line fire enough to reduce it a glancing blow. Nonetheless, the magic takes what effect it can, and more shadow seems to bleed away from the sleeper. Its struggle is now entirely to escape, and despite its injuries, it does seem to be steadily drawing the two of them towards the cave mouth. There might be time for one more shot before Bran is forced to let go or risk braving the dark cave, but with the cave-sleeper having greater control of the grapple, there's a small chance it could try and interpose Bran between it and the bullet. 

Kiritsugu freezes, his mind suddenly filled with the thoughts of a ship drowning in blood, of strangling his own wife, of it being all for nothing.  

He does not know what to do. He should shoot, but he cares about Bran and there are so few people in this world. This is the very act of evil utilitarianism mocked him with. 

He lowers the gun. "Bran, get out there, it's not worth your life." 

If Bran doesn't run he will grab him. 

Bran seems to accept that, or he simply can't hold onto the cave-sleeper any longer. Regardless, he lets it go, and it quickly disappears into the inky darkness.

Bran falls onto his back, and just lays on the ground for a moment, breathing heavily. He's back up on his feet fairly soon, though. He hustles back to Kiritsugu's hiding spot, and as he approaches Kiritsugu it becomes clear that he's covered in scrapes and cuts, each slowly leaking boiling golden ichor.

"I consider that a great success, my friend Emiya. I believe that it will be forced to recover for a great while, perhaps even all the rest of summer. Thank you!"

Kiritsugu bows. "I am glad I was of service. Is it safe to return?" 

He is impressed at Bran's physical prowess. "How do you survive that darkness against your body?" 

Heading back is going to take the same amount of time though, isn't it.

"It is hard to describe, I have not found any words for it yet. I know that I have a peculiar relationship with the hearth, how it is a part of me, or I a part of it. I focused on that connection, fed it with my mind and my breath, drew on it. I've done this a few times, including all the other times I've had the misfortune of wrestling the beast."

As he speaks, his wounds begin to close, his blood boiling away and leaving no stains on his skin or his now damaged clothes. He's still glowing a bit, but its steadily fading as well.

"That is... astonishingly impressive," Kiritsugu says. "I am glad that I was able to be of help. Do you think you'll be able to make the journey back, or should we find a place to hide for the night? 

He's still not quite sure how time moves here, and doesn't know if there are other dangers other than the cave bear when darkness falls. 

Bran smiles warmly at the praise. “I believe I can make it back home, if you can. If we start now, we should be able to get back sometime in the afternoon. I will recover much more quickly by the hearth, and I imagine you will as well.”

He seems to move easily enough, though with Kiritsugu’s experience he can tell he is suppressing some amount of pain.

"I agree." He reaches out hesitantly, before pulling his hands back. "If you are injured, I may be able to help some. I cannot change something as deep rooted as your mother's wrists, but I can help with minor cuts or bruises."

Briefly, Bran seems surprised, though his expression is thankful again just a second later. “I would appreciate it.”

Since his magic seems to work here, he should be able to knit together a few scrapes and such, binding the skin together in a way that does not quite erase that there was an injury, but should at least ease pain. 

He steps back after, looking slightly awkward. It's an obvious application of cutting and binding he doesn't use much, which is more than a little embarrassing.