That's pretty common. What mail program are you using? Your laptop's mail program is probably set to:
1) Use POP3 to retrieve email and is set to remove the email from the server immediately after it downloads to the laptop.
2) It's using IMAP, but you have mail routing rules that are moving emails out of the Inbox (an online folder) to an offline folder that is only accessible on your laptop.
In both cases, once that email is no longer in the inbox on the server, your Apple devices will no longer see them because they're using a mail protocol that syncs their inbox contents with what's currently on the server they fetched the email. If you already downloaded a copy of that email, then your iOS device will delete your local copy as soon as it realizes the original copy is gone from the server.
You can fix this revising the setup of your laptop's mail service. If it's POP3, you'll need to remove the account and set it up again as an IMAP account (the default option). If it's simply moving them out of the inbox, you just need to revise the mail routing rules that are causing the emails to move to an offline folder or designate the target folder as an online folder so the emails will be stored online and be available to all connected devices.
If you're dealing with the POP3 scenario, you could also change the amount of time that your mail program will wait before clearing the email out of the server's inbox after downloading. I don't think there's a limit to the number of days you could designate. That would be a messy workaround, but you'd at least have key emails for X number of days before they vanish.