I use an ancient copy of Outlook to download my e-mails onto two computers - my laptop and my old fashioned ancient tower unit which fires its output to a genuine CRT device!
My laptop e-mail sustem is set so it downloads any e-mails awaiting my attention but DOES NOT erase them from the server. It remembers which e-mails it has downloaded so if I do a fresh e-mail enquiry the old e-mails, still on the server are not downloaded again. My "mainframe" as I like to call it will then download all e-mails on the server when requested and delete them from this source. This is to ensure that every e-mail I have been sent will always be found in one place. It works for me.
P.P.