I think you're wrong when you compare transitions in mobile to the animations to the early 2000's flash "episode".
These animations where more like "we can do it", while the transitions in mobile apps actually serve a good purpose.
If a link is clicked that navigates horizontally (e.g. a "next page" link) a nice and well-known slide-in animation from the right (at least in left-to-right reading countries) supports the overall user experience by
- providing an additional hint on where in the app he is
- shadowing possible loading times
If a link leads to a different application, it uses a "new window" transition, wich helps the user to understand that he is now somewhere else.
This can also work for new windows in the same app. On android, this animation kind of indicates that the user can return to the previous screen by hitting the "back" button, what isn't neccesary true for paging or tabs.
Another example for a supporting animation can be used in combination with the Responsive Disclosure pattern.
Depending on the data that‘s about to appear, a decent animation can support the mental connection of the different data sets.
Stick to the platform standards and keep it consistent.