I will suggest below that the running total of constellation $18,18,12,12$ to stay pretty close to $\frac23$ of the running total for $12,18,12,18.$ That should be easy to check up to some limit.
Consider a particular constellation of $k$ integers considered to be roughly of magnitude $x$. For all large enough primes the members are in $k$ distinct congruence classes $\bmod p.$ But for small primes they are not. Knowing the data for all primes should give a conjectured density $\frac{c} {\log^kn} $ Depending on those small primes.
Before giving a specific example, I pause to say that there is no proof that this frequency is actually occurs, or even that the pattern happens infinitely often (if feasible at all), however there is strong heuristic and computational support.
Consider the constellation $12,18,12,18$ belonging to five integers $a,a+12,a+30,a+42,a+60.$ For $p>7,$ they chance that none is a multiple of $p$ is $\frac{p-5}{p}$ (provided $p \ll x$). And it is $\frac12,\frac23,\frac35,\frac37$ for $p=2,3,5,7.$ That should allow a computation of the probability.
Now consider the constellation $18,18,12,12$ belonging to $a,a+18,a+36,a+48,a+60.$ Again the probability that none divides by a given small prime $p >7$ is $\frac{p-5}{p}$. And again $\frac12,\frac23,\frac37$ for $p=2,3,7.$ But it is $\frac25 for $p=5$.
Accordingly, I would expect what I said at the top.