I have some ai files with <Mesh> layers and when I import the files in After Effects those layers are only shown with outlines. What would be a good way to deal with this and get some type of a working solution to use in After Effects for these cases?
There is no such thing. It's an unsupported feature in AE. If any animation or further editing is planned, you have to use AE's native mesh-like tools like the Mesh Warp effect or the Puppet Tool.