Three-time gold medallist Dujardin withdrew from the Paris Games on Tuesday after a video emerged of her repeatedly whipping the animal on the legs.
Dujardin, the joint most-decorated British female Olympian, said she had made an "error of judgement" and was suspended by the International Federation for Equestrian Sports (FEI).
The 39-year-old embarked on her dressage career under the tutelage of Hester, who part-owned Valegro, the horse on which she secured team and individual gold at the London 2012 Games.
Hester, taking part in his seventh Olympics, won gold alongside Dujardin in the team dressage event in London, claiming a silver in Rio and bronze in Tokyo.
The 57-year-old, who will compete for Team GB next week, is one of 10 signatories to a statement from the International Dressage Riders Club board addressing the incident involving Dujardin.
The group said they "universally condemn" the actions of Dujardin.
"The board members are resolute that equine welfare must always be placed uppermost and at all times," it said.
The letter added: "The IDRC board supports the actions taken by the FEI, the British Equestrian Federation and British Dressage to provisionally suspend Ms Dujardin.
"It notes that Ms Dujardin has stated she has no excuse for her actions and has also expressed her shame and regret."
A day after horse welfare charity Brooke dropped Dujardin as an ambassador, the London International Horse Show followed suit.
Two of Dujardin's sponsors confirmed they had ended their arrangements with her -- riding helmets manufacturer Charles Owen and Fairfax Saddles.
Dujardin could have become Britain's most decorated woman Olympian in Paris. A medal of any colour would have taken her clear of cyclist Laura Kenny, with whom she is currently tied on six medals.
She was due to compete in both the individual dressage and the team event alongside Hester and world champion Lottie Fry at the Paris Games.
Team GB confirmed on Wednesday that Dujardin had been replaced in the team by Becky Moody.
The dressage competition at Paris 2024 begins on July 30 at the Chateau de Versailles.
