Gmail emails sent on iPad don't show up among Sent items on MacBook Pro

Dear Apple Folks,

If I send an email from my Gmail account (via my Mail client) on my iPad (using iOS 9, but this also happened long before upgrading), the sent item doesn't show up among the Sent items in Mail on my MacBook Pro (running 10.9 at present). Why? The item is there in the Sent items if I go to the web-based version of Gmail, but frankly I'd like to have the Sent folders consistent across my iPad and MacBook Pro (and my iMac, for that matter). Is there a way to ensure the Sent folders across the Apple ecosystem are actually synced regardless of what machine sends the message? If the Inbox can be synced without issue, why can't the Sent box? Is there a setting I need to check in Mail on my MBP?