"round of emails" is correct and usable in written English.
You can use it to refer to a series of emails sent or received over a period of time, usually involving several people. For example, "We've been going back and forth in a round of emails trying to resolve the issue."
After the event, Mr. Mellul even included a Secret Service agent in his round of emails thanking those with whom he had worked.
In summer 2015, another round of emails was unleashed, apologizing for the continued delay and again offering refunds.
For example, in the last round of emails Start Mobile got letting them know why their app was rejected, it was just signed "iPhone Developer Program".
But after that first round of emails, we realized some of them aren't.
Wait a day or two for the responses, then take the lowest offer you get back and send another round of emails.
I sent out another round of emails, this time grounding my interest more specifically in the desire to "awaken dormant abilities in my mind and psyche".
The anti-Clinton forces consider it a victory to see yet another round of email stories written, of course.
I can't count the days when I've thought that if I have to vet one more round of email, I'm going to shoot my computer.
This first round of email messages will be followed by more efforts to recapture consumers who may have given up on the process while HealthCare.gov remains unreliable, Julie Bataille, a spokeswoman for the Centers for Medicare and Medicaid Services, said during a conference call with reporters Tuesday.
For those estimates not reaching consensus, the plan was for results to be electronically distributed to the panel, virtual discussion allowed, and a second round of email questionnaires sent.
After two rounds of emails to Mike Bevens of Brittany Ferries, his firm position was: "Unfortunately we have to draw a line somewhere with our support for charities and our focus has been set on local ones.
