"I leave Paris with a bitter feeling. Yesterday I was booed after only being respectful of my opponent's willingness not to shake hands after the game. Me and Elina have always respected each other after a tough match and to have to leave like that was the worst of all yesterday," Kasatkina wrote on Twitter.
Like many other Russian athletes, she has moved outside the country and has Barcelona as her base.
Former world number three Svitolina is perhaps the most active of the Ukrainians protesting the war, visiting Volodomyr Zelenskyy at the presidential palace in Kyiv, and consistently refusing to take Russian and Belarusian players by the hand after matches.
"Grateful for her stance"
Yesterday, Kasatkina, who called the war a "large-scale nightmare", gave a thumbs up to Svitolina, who responded in the same way.
"I am grateful for her stance and she is a brave person who dares to say it publicly. Not many people dare to do that," Svitolina said.
The crowd, who are loud and live up to the matches, have taken Svitolilna to their hearts after French singles players are eliminated from both the women's and men's sides.
Svitolina is married to the French legend Gael Monfils, who sat in her dressing room yesterday.
The war in Ukraine has come to affect the tournament in Paris. Since a couple of matches, Belarusian world number two Aryna Sabalenka refuses to participate in the post-match press conferences because she does not feel safe.