I don't think they would be dwarf planets, but something else.
The International Astronomical Union (IAU) defined in August 2006 that, in the Solar System, a planet is a celestial body that:
1 is in orbit around the Sun,
2 has sufficient mass to assume hydrostatic equilibrium (a nearly round shape), and
3 has "cleared the neighbourhood" around its orbit.
A dwarf planet must meet 1 & 2. Are Jupiter's smaller moons round?
Jupiter has rings, so any planet would have to have cleared the rings around their orbit. I think that applies to the Galilean moons. Juno orbits outside the solar plane, so I'm not sure if that is a rule for a planet or not.
Of Jupiter's moons, eight are regular satellites with prograd and nearly circular orbits that are not greatly inclined with respect to Jupiter's equatorial plane. The Galilean satellites are nearly spherical in shape due to their planetary mass, and are just massive enough that they would be considered major planets if they were in direct orbit around the Sun.
The largest ring of Jupiter is just about 129 000 km of radius. The nearest Galilean moon to jupiter has a semi-major axis of 421 800 km, so the rings aren't in any of their orbital neighborhoods.
Beside, the largest of them, Ganymede, is more massive than Mercury.
But you're right that not all the moons would be either planets or dwarf planets, many would be asteroids.