Sudeikis filed a custody petition, but Wilde motioned to dismiss the action, arguing that the children have spent most of the last four years either in London or Los Angeles, attending school in both cities.
Wilde was served the court documents in April this year while she was speaking about her new movie Don't Worry Darling on stage at CinemaCon in Las Vegas.
She was in the middle of a presentation to industry executives when she was handed an envelope by a court process server in a moment that sparked furious speculation among onlookers about the envelope's contents.
In the court motion, Wilde referred to the incident for the first time, calling out Sudeikis for his "outrageous legal tactics" and delivering the documents "mid-speech".
"Jason's actions were clearly intended to threaten me and catch me off guard. He could have served me discreetly, but instead he chose to serve me in the most aggressive manner possible," she said.
"The fact that Jason would embarrass me professionally and put our personal conflict on public display in this manner is extremely contrary to our children's best interests.
"Since Jason has made it clear that we will not be able to work this out for our children's sake outside of the court system, I filed a petition for custody in Los Angeles."
Sudeikis claimed he had requested the service take place at Heathrow Airport rather than at her boyfriend Styles' home.
"I did not want service to take place at the home of Olivia's current partner because Otis and Daisy might be present. I did not want service to take place at the children's school because parents might be present."
But a series of "logistical mishaps" meant the server was unable to serve Wilde at the intended location that day, he said.
He claimed he only learned of the public incident when it made headlines and said he was "deeply upset" by it.
Sudeikis said he later found out that Wilde's whereabouts were tracked down thanks to a tweet speculating her appearance at CinemaCon.
