I am looking for a way to sync passwords between Apple Keychain and LastPass or 1Password1. Apple Keychans works quite well only if you are on Safari. The others work well if you are not on Safari Mobile, so I am looking for a solution that is more flexible, which makes me think that some kind of sync would be needed.
Perhaps you mean 1Password by AgileBits? If so I can answer that as a happy user of 1Password that you can't. Migrating or syncing passwords from/with the keychain is as far as I know, not strictly possible.
But 1Password (at least) is designed as a replacement for the Apple Keychain. the migration is not without its pain points (manually moving passwords over wholesale, or a few at a time as they pop up is not fun) but once you have all your passwords moved over your need for Keychain is gone.
Also, you can get an iOS version of 1Password as well which syncs with the desktop versions (either through the cloud or locally within your WLAN) and ties in nicely with Mobile Safari and other apps on your iDevice.