I have a mesh M, that is made up of triangles. This mesh will then move in a straight line L that you can represent as a single vector. I need to detect if this sweeping mesh will collide with a static line segment, preferably in an a priori (continuous) way. Here is a MRE:

pub struct Triangle {
    pub a: Vec3,
    pub b: Vec3,
    pub c: Vec3,
}

pub struct Line {
    pub a: Vec3,
    pub b: Vec3,
}

pub struct Mesh(Vec<Triangle>);

fn collides(mesh: Mesh, path: Vec3, line: Line) -> bool {
    // ...
}

And to better help visualize it, here is an interactive example:
https://www.desmos.com/3d/awxbt16pqj
I need to detect if the gray line will collide with the moving (red) cube.

All resources online are either mesh/mesh collision detection or use discrete simulation steps. I would like this to be as fast as possible (as it will be run hundreds of thousands of times on the gpu), and I would like to use a continuous solution. It is also important to mention that I do not need to know about where the 2 items will collide, just if they will.