The Interwoven Story of The Brotherhood, Arya, The Hound and Lady Stoneheart (Spoilers Extended)

Background

The sequence of events in ASOS in the Riverlands is masterful. Not really anything new, but I just love the fact that all of this happened in such a short period of time.

#1 The Brotherhood catches Arya Stark to ransom to Lady Catelyn/King Robb at the Twins

After fleeing Harrenhal Arya is caught by the Brotherhood:

"Little one," Greenbeard answered, "a peasant may skin a common squirrel for his pot, but if he finds a gold squirrel in his tree he takes it to his lord, or he will wish he did."

"I'm not a squirrel," Arya insisted."

"You are." Greenbeard laughed. "A little gold squirrel who's off to see the lightning lord, whether she wills it or not. He'll know what's to be done with you. I'll wager he sends you back to your lady mother, just as you wish." -ASOS, Arya III

and:

"Your brother may be gone," said Thoros. "Your mother as well. I did not see them in the flames. This wedding the old one spoke of, a wedding on the Twins . . . she has her own ways of knowing things, that one. The weirwoods whisper in her ear when she sleeps. If she says your mother is gone to the Twins . . ." -ASOS, Arya VIII

#2 The Brotherhood catches The Hound

One thing that is interesting here is that Cat also set Jaime free and this chapter (Arya V occurs a few chapters after Jaime III, (which also occurs in the area):

"Have they caught the Kingslayer?" Gendry wanted to know.

Down in the square, a thrown stone caught the captive on the cheek, turning his head. Not the Kingslayer, Arya thought, when she saw his face. The gods had heard her prayers after all. -ASOS, Arya V

#3 The Brotherhood frees The Hound

After defeating/killing Lord Beric in his trial by combat, Sandor is set free:

Lord Beric shook his head. "Clegane won his life beneath the hollow hill. I will not rob him of it."

"My lord is wise," Thoros told the others. "Brothers, a trial by battle is a holy thing. You heard me ask R'hllor to take a hand, and you saw his fiery finger snap Lord Beric's sword, just as he was about to make an end of it. The Lord of Light is not yet done with Joffrey's Hound, it would seem." -ASOS, Arya VII

#4 The Hound steals Arya Stark from The Brotherhood

Arya tries to run away and she thinks she is caught by a member of the BwB:

"You're hurting me," she said, twisting in his grasp. "Let go, I was going to go back, I . . ."

"Back?" Sandor Clegane's laughter was iron scraping over stone. "Bugger that, wolf girl. You're mine." He needed only one hand to yank her off her feet and drag her kicking toward his waiting horse. The cold rain lashed them both and washed away her shouts, and all that Arya could think of was the question he had asked her. Do you know what dogs do to wolves? -ASOS, Arya VIII

#5 The Hound wants to ransom Arya to Lady Catelyn/King Robb at the Twins

Similar to the BwB, the Hound intends the same thing:

"Stupid blind little wolf bitch." His voice was rough and hard as an iron rasp. "Bugger Joffrey, bugger the queen, and bugger that twisted little gargoyle she calls a brother. I'm done with their city, done with their Kingsguard, done with Lannisters. What's a dog to do with lions, I ask you?" He reached for his waterskin, took a long pull. As he wiped his mouth, he offered the skin to Arya and said, "The river was the Trident, girl. The Trident, not the Blackwater. Make the map in your head, if you can. On the morrow we should reach the kingsroad. We'll make good time after that, straight up to the Twins. It's going to be me who hands you over to that mother of yours. Not the noble lightning lord or that flaming fraud of a priest, the monster." He grinned at the look on her face. "You think your outlaw friends are the only ones can smell a ransom? Dondarrion took my gold, so I took you. You're worth twice what they stole from me, I'd say. Maybe even more if I sold you back to the Lannisters like you fear, but I won't. Even a dog gets tired of being kicked. If this Young Wolf has the wits the gods gave a toad, he'll make me a lordling and beg me to enter his service. He needs me, though he may not know it yet. Maybe I'll even kill Gregor for him, he'd like that." -ASOS, Arya IX

#6 The Hound "saves" Arya from running into the Red Wedding and dying

Arya tries to run into the Red Wedding to save her family:

Arya spun away from him and darted for the gate. The portcullis was coming down, but slowly. I have to run faster. The mud slowed her, though, and then the water. Run fast as a wolf. The drawbridge had begun to lift, the water running off it in a sheet, the mud falling in heavy clots. Faster. She heard loud splashing and looked back to see Stranger pounding after her, sending up gouts of water with every stride. She saw the longaxe too, still wet with blood and brains. And Arya ran. Not for her brother now, not even for her mother, but for herself. She ran faster than she had ever run before, her head down and her feet churning up the river, she ran from him as Mycah must have run. His axe took her in the back of the head. -ASOS, Arya XI

and:

"Did you hit her with an axe too?"

"I hit you with the flat of the axe, you stupid little bitch. If I'd hit you with the blade there'd still be chunks of your head floating down the Green Fork. Now shut your bloody mouth. If I had any sense I'd give you to the silent sisters. They cut the tongues out of girls who talk too much." -ASOS, Arya XII

#7 Arya (Nymeria) drags Cat's body from the Green Fork while she is with The Hound

During one of her wolf dreams:

She splashed noisily through the shallows and threw herself into the deeper water, her legs churning. The current was strong but she was stronger. She swam, following her nose. The river smells were rich and wet, but those were not the smells that pulled her. She paddled after the sharp red whisper of cold blood, the sweet cloying stench of death. She chased them as she had often chased a red deer through the trees, and in the end she ran them down, and her jaw closed around a pale white arm. She shook it to make it move, but there was only death and blood in her mouth. By now she was tiring, and it was all she could do to pull the body back to shore. As she dragged it up the muddy bank, one of her little brothers came prowling, his tongue lolling from his mouth. She had to snarl to drive him off, or else he would have fed. Only then did she stop to shake the water from her fur. The white thing lay facedown in the mud, her dead flesh wrinkled and pale, cold blood trickling from her throat. Rise, she thought. Rise and eat and run with us. -ASOS, Arya XII

#8 The Brotherhood resurrects Cat and inform her that her daughter is alive and they lost her

"She is," said Thoros of Myr. "The Freys slashed her throat from ear to ear. When we found her by the river she was three days dead. Harwin begged me to give her the kiss of life, but it had been too long. I would not do it, so Lord Beric put his lips to hers instead, and the flame of life passed from him to her. And . . . she rose. May the Lord of Light protect us. She rose." -AFFC, Brienne VIII

#9 Lady Stoneheart has the Brotherhood interviewing Freys about The Hound/Arya's whereabouts before killing them

“The Red Wedding?” Merrett’s skull felt as if it were about to split, but he did his best to recall. There had been so much confusion, but surely someone would have mentioned Joffrey’s dog sniffing round the Twins. “He wasn’t in the castle. Not at the main feast … he might have been at the bastard feast, or in the camps, but … no, someone would have said …”

“He would have had a child with him,” said the singer. “A skinny girl, about ten. Or perhaps a boy the same age.”

“I don’t think so,” said Merrett. “Not that I knew.”

“No? Ah, that’s a pity. Well, up you go.” -ASOS, Epilogue

#10 The Brotherhood/Lady Stoneheart start tracking down the different people/groups that have interacted with Arya/The Hound

This is somewhat in tandem with #9:

The outlaw gave him an encouraging smile. “Well, as it happens, we’re looking for a dog that ran away.”

“A dog?” Merrett was lost. “What kind of dog?”

“He answers to the name Sandor Clegane. Thoros says he was making for the Twins. We found the ferrymen who took him across the Trident, and the poor sod he robbed on the kingsroad. Did you see him at the wedding, perchance?”

“The Red Wedding?” Merrett’s skull felt as if it were about to split, but he did his best to recall. There had been so much confusion, but surely someone would have mentioned Joffrey’s dog sniffing round the Twins. “He wasn’t in the castle. Not at the main feast … he might have been at the bastard feast, or in the camps, but … no, someone would have said …”
“He would have had a child with him,” said the singer. “A skinny girl, about ten. Or perhaps a boy the same age.”

“I don’t think so,” said Merrett. “Not that I knew.”

“No? Ah, that’s a pity. Well, up you go.” -ASOS, Epilogue

If interested: Lady Stoneheart: The Culmination of Numerous Riverland Plotlines

TLDR: Nothing new, just an example of how GRRM does a great job of interweaving different stories together and also that Lady Stoneheart isn't solely focused on killing. She knows her daughter is alive and has been actively searching for her.