You can download and keep older messages in the unified inbox folders by using Fetch more messages in the three-dots overflow menu of the start screen. For other folders, you can long press the folder in the folder list of the account (tap on the account name in the navigation menu = left side menu). When you long press on a parent folder, there will be a subfolders option with a menu item to fetch more messages for all child folders. Please read the remark in the confirmation dialog box.
Note that starred (favorite) messages will be kept on your device “forever”.
Instead of downloading many messages to your device, consider searching for messages on the email server.
+Longer explanation: the app has a sync and a keep window per folder. The Fetch more messages menu item is, in fact, a mini wizard to change the keep window of one or more folders at the same time, and to initiate a corresponding synchronization operation. If you long press on a folder in the folder list and select Edit properties, you can change the sync and keep window of the folder. The sync window determines which messages will be checked with a standard sync operation (default 7 days), and the keep window determines how long messages will be kept on the device (default 30 days). If you sync another time within 30 seconds, all messages in the keep window will be checked, which is useful when you have been moving around messages with another email client, for example on your desktop computer. It is mostly pointless to check all messages all the time, especially considering that this will use battery power, which is why there is a separate sync and keep window, and why the app checks messages younger than a week only.
(192) How can I resolve ‘Couldn’t connect to host, port: …; timeout …;’ ?