The phrase "a constant round of" is correct and usable in written English.
It can be used to describe a repetitive or ongoing series of events or activities.
Example: "Her life had become a constant round of meetings, deadlines, and social obligations."
Alternatives: "an endless cycle of" or "a never-ending series of".
These were heady days for Mary, a constant round of parties and country-house weekends.
Throughout 1977, McLaren reacted with lightning speed to a constant round of scandal, business upsets, and personal crises.
For Mr. Salzberg and his four nurses and office managers, life became a constant round of giving medicine, taking temperatures, talking to parents.
Where and how you worshipped mattered, too, and there was, above all, a constant round of teas, receptions, dinners, parties, balls and cotillions.
Robert Evans's diaries for the early 1830s record a constant round of comings and goings, with aunts and uncles descending periodically just as they do in The Mill on the Floss.
When he tries to close down Victorian-era hospitals and wards that have been mistreating elderly patients, he becomes "immersed in a constant round of demonstrations and petitions fighting to 'save' clapped-out institutions all over the country".
To be fair, he doesn't have a PA, and he does have a lot of commitments, such as his constant round of speaking engagements (more than 100 in the past year), his appearances as the token Rightie on Question Time, and his bold and elegantly written columns for the Spectator and Daily Telegraph (which he also used to edit, before Maggie took over his life).
Instead I can still feel it in the air, there was an expectancy, there was hope and there was an abundance of faith expressed, especially in the constant round of performances of Nativity plays in our churches and sometimes in the lounges of the local hotels of Montego Bay.
An army of lithe young girls has arrived from New Zealand, hoping to earn some euros as nannies during the constant round of America's Cup parties.
Finally my cousins showed us the side of Pakistan that is missed by the headlines of terrorist training camps and religious extremists: the constant round of parties, dances – and, of course, glowsticks.
