Dirichlet's theorem shows that, for any fixed prime integer a, "big prime numbers mod a" are uniformly distributed between 1 and a-1. If we similarly pick different prime integers b,c,..., are these uniform distributions independent of each other?
Yes (asymptotically of course). The prime number theorem for arithmetic progressions tells us that primes are (asymptotically) uniformly distributed in the $\phi(m)$ reduced residue classes modulo $m$ for any integer $m$, even composite $m$. You can quickly convince yourself that, if $a$ and $b$ are primes, the independence of the uniform distribution of primes modulo $a$ and modulo $b$ is exactly the same thing as the uniform distribution of primes modulo $ab$. The same holds no matter how (finitely) many primes $a,b,c,...$ you use.