They don't have to be downloaded. Downloading will occur if you utilize POP protocol, but with IMAP they normally will not be (except as HTML or text for viewing purposes). Several years ago I decided that I really didn't need most messages downloaded anymore and so changed to IMAP. Since I have two big email providers both backing up (so there are two separate backups of each message), why do I need to? But if you really want some messages downloaded you can do it on a folder by folder basis (with Thunderbird). This might make sense on the highest-value messages.
when I “get mail” with Thubderbird, I “get” and yes “download it” into Thunderbird - all of it, and have Thunderbird’s setup arranged to instruct the ISP’s Email system to delete everything when it is passed to Thunderbird.
With junk mail filters and message filters in Thunderbird I can preassign just about any Email I get into a relevant folder I have (dozens) in Thunderbird.
First off I prefer to NOT sit on the web while I read/review/process my Email - I’m offline once my Email is in Thunderbird until I want to send/reply, which I do in Drafts and then send all the drafts at once.
I never, never ever exceed any of my ISP’s limits, no matter how many images and how big the images and how much data is sent to me in Email - nothing stays on the ISP’s website for more than 12 hours.
My active Email in Thunderbird is not stored on my PC’s hard drive; in fact ALL my own “stuff” is on an external USB connected hard drive and NOTHING of my own stuff is on my PC’s hard drive, and both drives have their own back ups.
When I travel, my Thunderbird Email travels with me on my remote, USB connected hard drive.
Nothing is either lost, or resident on my ISP for long.
Why play around with the ISP’s sloppy and cumbersome web-based Email system pages when having Thunderbird can simply use their system as a slave, give me my Email in my own computing environment and bypass everything else the ISP wants to share with me on their web pages.
In fact, while I have a major ISP with high speed fiber optic broadband, I absolutely never have any occasion or need to visit any of their web pages.
They provide two services to me - deliver TO ME essential Internet service, and deliver TO ME ON MY PC my Email.