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Looking that the source code, it seems that these are implemented using shaders. Those are a new thing in Flutter, so I'd try to go that road. You probably need a CustomPaint with a CustomPainter, painting a Rect with a Paint using that Shader, feeding it all input parameters. I've absolutely no experience with shaders in Flutter, but that's what I'd try for something simple like the fog. For the birds, you'd probably need to implement a lot of 3D geometry yourself.
It uses https://threejs.org/. Unless you intend to port 3js to Dart, it's a no go.