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The manually forwarded emails should be fine as they would be coming from your email address (so long as your company's email auth. is up to scratch).
It's the auto-forwards that can be a bollox and it would be the same no matter which infrastructure you use. I can't see that Expensify are using DMARC on their outbound (at least not on any of the usual domains - expensify.com, e.expensify.com, email.expensify.com, info.expensify.com) so if they aren't, it's unlikely they've set it up on the inbound so you're probably fine.
Gmail uses Filters to do auto forwards. There are a few help pages if you Google it.
I wouldn't rely on it to deliver my email though because of what I have said above.
Essentially it is a means of you "spoofing" the domain of the person who originally sent you the email, albeit for legitimate reasons. This is being clamped down on more and more and if someone implements anti spoofing and doesn't care about protecting forwarded messages where they can (why would they, their email got to where it was supposed to go), the Ban Hammer comes out.
Cheers, this sounds more complicated than I thought. I use Expensify for expenses, emailed receipts (hotel, flights, etc) are forwarded onto them manually by me or some are forwarded by rules I have set up in Outlook (work email). I assumed I'd be able to set up similar rules in Gmail but seems it's not quite the same.