If you want to use the empty object as driver of the material, the material should look like this:

• Texture Coordinates: use the Object output, and select the empty object in the object selector. Make sure the empty is really close to the mesh you want to affect.
• Gradient texture: set to Spherical
• Color ramps: no need for two of them, one for coloring is enough. Tip: if you set the color selector to HSV, you can manually write down values >1, which allows you as in my example to have the yellow color stop to emit light at 10 strength while the orange is at 1. Allowing a more granular control of light emission.
• Principled shader: use the gradient texture in its Alpha channel if you want transparency.

I still recommend using a curve for animating the empty as Chris said though, if you only need it to go somewhat linearly from start to end with nothing more complex:

If you want to use the empty object as driver of the material, the material should look like this:

• Texture Coordinates: use the Object output, and select the empty object in the object selector. Make sure the empty is really close to the mesh you want to affect.
• Gradient texture: set to Spherical
• Color ramps: no need for two of them, one for coloring is enough. Tip: if you set the color selector to HSV, you can manually write down values >1, which allows you as in my example to have the yellow color stop to emit light at 10 strength while the orange is at 1. Allowing a more granular control of light emission.
• Principled shader: use the gradient texture in its Alpha channel if you want transparency.

I still recommend using a curve for animating the empty as Chris said though, if you only need it to go somewhat linearly from start to end with nothing more complex. Just add a curve, shape it how you want the object to move, then add a Follow Path constraint on the empty object.