This thread is a null issue
Considering its OP was in 2011, no kidding!
focusing on syncing data between mail clients isn't a 'thing'. Instead look into each mail client application (PC) or app (mobile device) tying into the email service involved (Gmail, Outlook, Yahoo Mail, AOL Mail, etc.).
Much depends on the individual's needs.
As noted, I did install the app,
FairEmail, named in the post that resurrected the thread. I've since paid for its Pro version.
I've looked at various Android mail apps over the years, but they all lacked something I needed/wanted. This one doesn't! It gives me nearly the same functionality I'm used to in my real mail client,
SeaMonkey, which I've used on my computers since its days as Netscape Communicator (and it's Thunderbird's predecessor).
I don't use Gmail or its app at all; the only mail that comes to that address is from Google, like Play Store purchase confirmations. I have SeaMonkey set up to handle Gmail, my private Earthlink address (I've had for 25+ years!), and all the mail for my three domains. In SeaMonkey, all those accounts are seen and accessible at once, and that was one main thing I was looking for in an Android app.
I use my main computer as the central location for all my mail, so anything received on my phone stays on its respective server until downloaded on it. I have mail I can still access with SeaMonkey from the '90s!