The former working royals were tailed by photographers on Wednesday night after Meghan accepted an award in New York.
They were filmed leaving the venue in a blacked-out SUV and were later seen climbing into a taxi cab before getting back into the SUV.
Harry's team described the "relentless" ordeal as a "near-catastrophic car chase".
In a statement, a spokesperson blamed a "ring of highly aggressive paparazzi" for the incident, but an eye-witness has since played it down.
They told SundayStar.co.uk: "It wasn't a high-speed chase at all.
"Harry and Meghan's taxi got blocked by a trash truck so it was just stopped in the middle of the road."
Slamming claims the "chase" was "high speed", they added: "It's Manhattan so the car was creeping through traffic anyway.
"The whole thing was more like a slow-speed crawl through New York before they got back into the black SUVs and continued their journey."
The New York Police Department told us: "On Tuesday evening, May 16, the NYPD assisted the private security team protecting the Duke and Duchess of Sussex.
"There were numerous photographers that made their transport challenging.
"The Duke and Duchess of Sussex arrived at their destination and there were no reported collisions, summonses, injuries or arrests in regard."