However, a suspension put Walker on the sidelines with the ride going to Craig Williams, who partnered the son of Sebring to victory in the Queen Elizabeth Stakes at Randwick before departing on a three-race international campaign.
Walker said Criterion had returned to Australian shores a tougher horse in the mind. "To me, from when I first sat on him to now, if anything, he's mentally a different horse.
"Physically, he's always been a beautiful animal and I'm just looking forward to riding him in a race.
"If everything comes off he could be one of the best stayers that I've ever ridden."
Walker galloped Criterion over 1600m, coming home the last 400m in 24.8secs, quicker than expected.
"The good thing about him he was really relaxed and wouldn't have blown a match out afterwards," Walker said. "He did the gallop stress-free and walked back nicely.
"Fingers crossed, toes crossed, everything crossed, I'm hoping he comes through every race with flying colours and he ends up running in all three races."
Criterion has drawn seven in the field of eight and is an $11 chance, with Kermadec the $2.90 favourite.
Meanwhile, five internationally-trained horses remain in the Caulfield Cup after third acceptances for the A$3 million handicap on Saturday week. A total of 38 horses have been paid up for at the latest declaration stage for the 2400m feature.
English horse Snow Sky, Japanese stayers Fame Game and Hokko Brave, Ascot Gold Cup-winner Trip To Paris and the Roger Charlton-trained Quest For More are the remaining Northern Hemisphere entries. Final acceptances are taken next Tuesday.
Trainer Chris Waller has confirmed he plans to have four runners in the Caulfield Cup and possibly three in the Cox Plate.
The stable's intended Caulfield Cup runners are Beaten Up, Grand Marshal, Royal Descent and Who Shot Thebarman.
"Glen Boss will partner Royal Descent, Jim Cassidy will be aboard Grand Marshal, Glyn Schofield is booked for Beaten Up and Blake Shinn will be on Who Shot Thebarman," Waller said.
"Our Cox Plate runners will be finalised after this Saturday's racing, however, it is our intention at this stage to run Kermadec, Winx and Preferment.
Arod, Gailo Chop and Highland Reel are the three international entries left in the Cox Plate. AAP