As with Donkey, Diddy, Rambi, and Winky, Expresso can be made to walk with the arrow buttons, and to run by holding them down with the Y button. However, due to a couple of cases that make him run all of the time, it's easy to think that walking isn't an option. Using Barrel Cannons (such as when emerging from the bonus area in Temple Tempest with Expresso) or jumping and flapping your wings will put the bird into an always-run state. As the ostrich is very fast, this can compromise control some, and increase your chances of moving into a ditch or an enemy.
To reliably disarm the bug, dismount, have your Kong move around until he stops auto-running, then remount. However, it'll be re-enabled every time you flap.
What's happening is that the game will continually set an "Auto move faster" flag (Bit 1 of 7E/169A for Donkey, $169C for Diddy) from the time you flap your wings in mid-air until you land. (This is done at BF/97B2 in USA Version 1.2.) Presumably, this is to let you fly faster and further without having to hold Y, as holding one button while pumping another with the same thumb is a pain in the ass (see Super Mario Bros. 3, Level 6-5). While the flag is no longer repeatedly set when Expresso is on the ground, it doesn't get cleared, either. It does with a Kong (albeit after a possible delay). That's why you'll need to dismount to allow walking.
The flag also gets set by jumping off of a rope, which is another kind of mid-air jump. (I don't yet grasp the reasoning behind this or the Barrel Cannons' setting. I'd have to study game behavior with it never being set.)
Regarding a fix, I'm too clueless of this game's internals to know where to put it. Because BF/97B2 sets the bit continually, I might be able to get away with continually clearing it shortly before the containing function is called, but that's cheap and risks affecting too much.
I discovered this bug circa July 4, 2006 or sooner, but inexplicably never typed up a post until this January. Google searches and looking at a few good webpages comes up empty, which makes me think that this one is so pervasive, players don't even realize they're being deprived of walking.