"I had a nice day. I felt like I moved on moving day. I hoped I could close the gap a little bit with the people teeing off now and a little bit later," said Lee, who shot a 68 to move to two under par.
Kyriacou, whose sensational hole-in-one stole the headlines on Friday, carded a 69 and shares 11th place on three under.
Conditions remained challenging on Saturday at Royal Porthcawl in Wales, and Japan's Miyu Yamashita maintained her lead despite stumbling to a two-over 74.
Yamashita is at nine under, but her advantage was narrowed to a single shot heading into Sunday's final round.
A Lim Kim of South Korea moved into second place at eight under, with American Andrea Lee third on seven under.
Australian Lee was an early starter alongside compatriot Grace Kim, the only other Aussie who had made the cut, whose 74 left her well adrift on four over.
"I missed putts on two and three, obviously 18," Lee said. "Had a lot of really great looks out there. I feel like the short ones, I'm over-reading them a little bit.Â
"It's kind of tough because you think the surface is not going to break, but sometimes it really does break. It's just the way links golf is. Sometimes it's hard to pick the breaks."
Kim produced one of the most thrilling finishes in major championship history last month to win the Evian Championship, and Lee said: "I think, for Grace, it's probably too close to her winning so far. She's probably still a little bit on the high.
"She's only had one week off and had to play the next two. It's probably maybe when we have two weeks off next week, she'll be able to really relax and soak it more in maybe."
As for her own chances on Sunday, Lee added: "I think tomorrow is going to be go as low as I can. I'm probably too far back."
After her roller-coaster round on Friday, Kyriacou, who made four birdies, said: "Yes, definitely not as exciting. I mean, it's not a bad thing. Less bogeys on the card, which was a goal today. I haven't really checked the leaderboard yet. I assume I moved up in this wind.
"I dropped one today. I dropped four yesterday, which was quite rough.
"I'm pretty sure I have to chase (on Sunday), foot on the gas. The same goals as the last couple days. I think I said this yesterday, but you just have to be brave and play with the elements out there."
A group of four tied for fourth place at six under included Japan's Minami Katsu and Rio Takeda, England's Charley Hull and American Megan Khang.
Yamashita and Takeda, who was alone in second place through two rounds, were the only members of the top 10 on the leaderboard to shoot over-par on Saturday. Takeda, Yamashita's playing partner, also was two over.
World No.1 Nelly Korda stumbled to a 74 to leave her at even par for the tournament in a tie for 36th place.
Defending champion Lydia Ko of New Zealand shot a two-under 70 and was also tied for 36th at even par.
With agencies
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