AFter it runs out of ices, what remains will closely resemble an asteroid. Chiron is an object that may have once been a glorious comet. It may extend 100 million kilometers or more, and flows straight out away from the Sun. The dust tail is perhaps 10 times shorter than the ion tail and is made up of cake-flour like particles that reflect the Sun's light. These dust grains follow their own orbits, and fight against the pull of the Sun's gravity and the push of the Sun's radiative pressure.
Comet dust is responsible for meteor showers when the Earth crosses a comet's path.
Image from A Cosmic Perspective, Bennett et al. Comets have the most eccentric orbits of any solar system object. Their orbits may take thousands of years or less than 100. Comets follow Kepler's Laws. Comets must come from somewhere.