It is a well-known fact that mass by itself is not conserved (since, for example, a particle can annihilate with its antiparticle). However, in classical physics, and as long as there is no physical process of annihilation/creation of matter with mass, we have an "approximate conservation law" for mass (i.e. outside the processes of nuclear physics or quantum mechanics, we have no major violations of "mass conservation" in non-relativistic/non-quantum physics).
Furthermore, in classical mechanics, Noether's theorem explains under what circumstances the other conservation laws appear. However, it does not seem possible to use Noether's theorem to deduce that under the type of restricted processes of non-nuclear physics mass must be conserved. I wonder whether this implies that the conservation of mass in certain processes cannot be deduced from more general principles, and we must simply accept it as an irreducible and unexplained empirical observation.
One possibility is to start from a Lagrangian density $\mathcal{L} = f(\rho,u^\alpha,x^\alpha)$ to obtain the energy-stress tensor for a free-preasure "fluid":
$$T^{\alpha\beta} = \rho u^\alpha u^\beta $$
then:
$$\frac{\partial T^{\alpha\beta}}{\partial x^\alpha} \approx \frac{1}{c}\frac{\partial(\rho c)}{\partial t} + \frac{\partial(\rho v_x)}{\partial x} + \dots + \frac{\partial(\rho v_z)}{\partial z} = 0$$
which is precisely the continuity equation that expresses the conservation of mass.
