Eventually the time comes, when Applejack has taken the time to reflect and apologize to her daughter, that Azalea is ready to come home.

1-2)
Azalea stood at the end of the gravel path up to her childhood home. She wondered if both her mothers were home today, it was a Sunday after all. Winona II, an old hound dog scents her and perks, milky eyes searching for the familiar smell's origin. The elderly dog rises from her spot on the porch, and howls her delight.
"Hi, old girl." says Azalea, approaching the dog and lighting her horn, using her magic to ruffle the fur on Winona II's chest. "Did you miss me?"
Black Quartz, who levitated a crystal-berry pie in the red aura of her own magic, approached and smiled. She watched Azalea crouch down to nuzzle the aged pet, muttering praise into the dog's floppy ear.
The happy moment is cut short when the screen door swings open, and a plump white mare stands in the doorway. "AZAAALEEEAAAAAA!" she wails, leaping over Winona II and barreling the pink mare over. "MY BABY GIRL HAS COME HOME!" Rarity smothers her daughter with kisses and nuzzles, blubbering nonsense into Azalea's neck.
"Hi, Mama." soothes Azalea. "Is Mom home? I want you two to meet somepony."
Rarity sniffs, manifesting a kerchief out of thin air and wiping her face. She looks confused for a moment before she notices Black Quartz, who stands off to the side, the pie still floating in her magic.
"Oh! You've brought a friend!" Rarity gets up off the ground and helps Azalea stand. "Let me get Applejack. One moment." Rarity trots over to the side of the house and bellows "APPLEJACK, DARLING! YOU'RE NEEDED!" in the general direction of the back shed.
Applejack emerges from the shed, looking a little miffed by Rarity's call. "Rarity, sugarcube, there's no need to holler like that-" The orange mare is unable to finish her sentence, shocked into silence as she comes around the house, and sees Azalea.
"Hi, Mom." says Azalea, suddenly shy. "This is Black Quartz, my girlfriend."
Applejack and Rarity are both unable to think of anything to say, and just stand there, side by side, grateful for their daughter to be home again.

3)
The rest of that day was spent together. They shared Blackie's crystal-berry pie and talked, getting to know one another again after years apart. Eventually, night falls and the family goes to bed. Applejack and Azalea are both unable to sleep, though, and independently find themselves sitting on the back porch, watching the nigh sky together from the comfort of a porch swing.
The pair sat in silence for a while, before Applejack broke the silence.
"I'm sorry, sugarcube." began Applejack. "I'm sorry that I couldn't see that you were strugglin'. I let Granny's passin' get in the way of what was really important, and I failed you."
"Its alright." said Azalea.
"Its not though!" said Applejack, frustration in her voice. Applejack turned to Azalea, bitter and shameful tears in her eyes. "I hurt you, and I hurt our family. Your mother and I almost fell apart because of what I did."
Azalea averted her gaze, staring at her hooves. Her throat was tight.
"I missed you, Azalea. I missed you more than I could stand" said Applejack. "I'm sorry that I made you feel like I didn't care about you."
The first tear slipped over Azalea, opening the dam for the rest. She did what any child would do, and leaned on her mother, pressing her face into Applejack's warm, hay-scented mane.
"I missed you too." Whimpered Azalea, her voice crumbling into a sob.
Applejack wrapped her forelimbs around her daughter, and let her sob and cry until there were no more tears to shed, and the sky was growing light.

4)
By the time the sun rose, Applejack and Azalea were still sitting on the porch. They had fallen asleep together on the porch, Azalea curled up at her mother's side like she was a foal again.

Rarity, a lifelong early bird, was shocked to wake up alone, and then discover her wife and daughter together. She woke them both and sent them to their beds to get a few hours of proper rest before the day began. She held Azalea back for a few moments, once again conjuring a kerchief again to dab at her eyes.
"I'm so happy to have you home again." begins Rarity. "Your siblings, and your mother and I have missed you so very much."
"I know, Mama." says Azalea, a fresh tear forming in her eye.
Rarity levitates her kerchief in the blue shimmer of her magic, dabbing at Azalea's eye in a way only mothers do.
"Now to bed with you!" she says, voice bright. "I demand that you sleep half the day and when you get up, we'll do brunch, hm?"
The white mare bats her daughter with the kerchief, shooing her up the stairs and into her room. She kisses Azalea on each cheek before leaving her, prancing off to the master bedroom to pester Applejack.

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A happy ending for everypony!!! though this is not the end of Azalea's story just yet!