If you transform a mesh you need to recompute the AABB from its transformed vertices. This can be unnecessarily expensive for broaphase culling (e.g. in physics or visibility). A simple way to conservatively transform a given AABB is this:
rnVector3 TransformedCenter = Transform * Bounds.Center; rnVector3 TransformedExtent = rnAbs( Transform.Rotation ) * Bounds.Extent;
Here the rotation part of the transform is a 3x3 matrix and rnAbs() builds the absolute value of each element of the matrix. This formula doesn't give you the best fit AABB like transforming each vertex, but it is conservative and will contain your transformed mesh (e.g. it can be larger than necessary and yield false positives). Note that I used the center/(half)extent form of the AABB here, but it is trivial to transform from/to the min/max form. I would also not worry about the performance if you need to transform between the two!
Conceptually this formula transforms the original AABB (which now becomes an OBB) and then builds its AABB.