Most likely it was a password stuffing attack. If they used the same password on multiple sites, there is a good chance one of those other sites was compromised and the attackers took the compromised credentials and tried them on other sites like Instagram. It could have been something more advanced like a stolen cookie, but usually the simplest explanation is most likely.

Always use a different password for each service, enable MFA where possible, and use a password vault like Bitwarden.

She’s just been through her junk email folder and found a “We’ve noticed a new login” email from instagram yesterday

The junk-ing security notices is so common.
A few months ago, my dad said "uh, I got some email from my bank, and now my credit card doesn't work".
The email was describing some problem with his account which would have been so much easy to fix before they cancelled the card.
Similarly, I lost a domain name because the registrar notifications for renewal ended up in my junk mail.

It's probably quite a significant issue. Companies can go "well we tried to contact you" and wash their hands.
Doesn't matter that they also spammed bullshit marketing emails from the same address that issues security/renewal notifications.
Doesn't matter that spam email has been such an issue it is near-impossible to host your own email server (and expect delivery) for a decade or so now.

Are the messages in her outbox? What some scammers do is they make a new account with the same name and profile picture, to then send messages that look like they're from a friend.

I remember reading about an Facebook phishing thing going around, but that involved getting the password and then spamming out messages. She could try and remember if she got any weird links that she logged in with. Sometimes the scammer sends the harmful link to the followers, so the handbag site might be the problem link

Her account was likely compromised. It wouldn't be possible to say how without more info. Could have been that her password was hacked (guessed), or could have been that her session was cookie hijacked where they wouldn't have had to exploit the password.

I don't know if IG has 2 factor authentication, but that could be something to look into. Also ensure the new password is not dictionary based. Use a password manager like bitwarden.

first of all i dont use instagram so i cannot guarantee what i have said. however i do have a fair bit of roblox experience and account stealing kinda happens all the time there.

the website sent out could steal session details. most sites have a "remember me" feature so that you dont have to login every time you visit it. the server does this by giving your computer a unique identifier after you log in, so now that identifier can be used instead of a username and password. the website could steal that identifier and thus impersonate you, without getting your username and password.

after doing this basically anything could happen, but most likely the account will be controlled by a bot to spread the ad further, eventually amassing some sort of botnet.

in theory stopping the bot from accessing your account could be as simple as logging out of all devices the moment you fall for it, but im not too sure what would happen if you do it any later than that.

i dont think a proxy actually exists. to pretend to be someone else you either make an account that has similar characteristics as the target or just steal the target's account directly. is it literally every one of the followers that are targeted, 100%, or is it just most of them? it could be some sort of website like what i said spreading through the network of followers.