No, its not random- its listed in the times guide. On our last trip in May they stopped around 10:30 am at Japan in World Showcase. Since it was still early, most people were still in Future World so there were no lines for the characters in Japan. We gave a CM our camera and he offered to follow us around from character to character and take our pics- like a personal photo shoot That's where the pic in my signature is from.
Here's the current schedule according to the Steve Soares site:
The Disney characters are "on holiday!" Meet your favorite Disney stars when their sightseeing bus makes a scheduled stop in one of the following locations:
Future World Pin Central: Monday-Saturday at 9:05am
Spaceship Earth: Sunday at 9:05 & 10:10am; Monday-Saturday at 10:10am
Future World To World Showcase Bridge: Daily at 11:15am
Japan: Daily at 12:20pm
Showcase Plaza: Daily at 2:05, 3:10 & 4:15pm
Length of show: 15 minutes
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