*Disclaimer* I'm against the idea Avril ever had a body double, and definitely don't believe she was replace, here's what I believe instead.
Pre Let Go Era: Avril liked to sing at smaller events, and seeing clips of her sing, her voice seems very folk/country inspired. This is her using her natural voice and showing good control over it, and shows that her voice wasn't "deep" like people who believe the Avril is dead theory believe.
Let Go b-sides 2001: Many b:sides were co-written with Avril, but some weren't, which is why it was released outside of a few promotions, but vocal wise it's similar to the Let Go era, except it seems to be more on par with Avril's natural tone.
Let Go- Avril's first major exposure, in live performances Avril pushes on her larynx to create a deeper huskier tone to seem more "rock". This isn't good for her voice and makes her seem less skilled, and out of breath. So her voice didn't change the way she used it did.
Under My Skin- I think after the death of her grandfather she chose a darker sound and theme as a way to mourn, and in many live performances in this era she sounds very vulnerable.
Best Damn Thing- This is why some people want to believe she was replaced, but I think after the darkness that was Under My Skin, Avril went all out with a bratty upbeat image in a way to overcompensate for her sadness. Her overall image here is much spunkier and upbeat, and I believe this is because she wanted to feel happy and such. Vocally I believed she was coached, and she sounds very bratty on this album, especially live, but that's really the album's theme.
Goodbye Lullaby- Avril wanted to show growth, and she wanted to do it with this album, but songs like What the Hell and Smile (still like them though), held her back. The album was mean to be more introspective, and I really like this album, but I feel it should've been organized better. This album was hit or miss in terms of live vocals.
Avril Lavigne- Light-hearted and in my opinion an album most like Let Go, especially Here's to Never Growing up which is a call back to Complicated when it comes to song structure, not sound. I think vocally this is also the closest to the Let Go era, except she does let a little bit more of her natural tone through, and uses her voice to a much better extent.
Head Above Water- Big jump, Avril dresses a lot more formal for this album and wears longer clothes, I believe it's because after she got lyme disease which is caused by ticks she believe longer clothes that covered more skin would protect her better, also because formal suits the album well, but I think lyme disease has something to do with it. Avril's best live vocals in terms of skill, her control has never been better, however because lyme disease her vocals haven't fully recovered. I still think this is the best era so far for live vocals, and I think in terms of her commercial career she's only got better.
That's my theory and below are my rankings-
My Rating Of Live Vocal Eras
1st- Head Above Water
2nd- Pre Let Go
3rd- Avril Lavigne
4th- Goodbye Lullaby
5th- Under My Skin
6th- Best Damn Thing
7- Let Go (My favorite Era, but the strain she was putting on her vocals live was not good, but she still sounded pretty good here.)