Let $A$ be the set of all multiples of the first $n$ primes. The asymptotic density of $A$ should be given by $\mu=1-\prod_{i=1}^n(1-1/p_i)$. Letting $a_k$ be the $k$th element of $A$, the function $a_k\cdot\mu-k$ is periodic. For instance with $n=3$ (so $\mu=\frac{11}{15}$) it looks like:
$\hskip 1in$ 
With larger $n$ though (below $n=10$), things become more interesting:
$\hskip 1in$ 
With[{aa = Prime@Range@10, range = 1*10^5},
With[{bb = DeleteDuplicates@ Sort@Flatten@
Table[Select[Range@(range), Mod[#, aa[[nn]]] == 0 &], {nn, Length@aa}]},
ListLogLinearPlot[{(1 - Product[(1 - 1/Prime@j), {j, 1, Length@aa}])
bb - Range@Length@bb}, Joined -> True]]]
What is going on here, and why are the oscillations so regular?
NOTE
As noted by Daniel Fischer here, the sequence $a_k⋅μ−k$ is always eventually periodic (for $n=10$ its period is $5447823150$, so the plot for $k⩽10^5$ can't reveal the periodicity, and I would need to plot for $k⩽10^{10}$ or so to see it). I am referring then to a finer-grained regularity, probably best seen on a log scale, on its way to uniform periodicy.
Added
Standard plot & log plot for period $2$, $n=6$ as per Barry Cipra's suggestion.
This is really just a comment, but it's too long to post as such. Also, to a certain extent, it does address the OP's question, "what's going on here?"
The function $f(k)=\mu a_k-k$, as the OP noted, is periodic. Its period is $$N=(2\cdot3\cdots p_n)-(1\cdot2\cdots(p-1))$$
(E.g., for $n=3$ the period is $N=(2\cdot3\cdot5)-(1\cdot2\cdot4)=22$, as can be seen in the OP's top graph.) Because of the periodicity, using a logarithmic scale to graph $f$ is arguably the "wrong" thing to do: For large $k$, the graph will simply appear as a big smudge, since the periodic interval gets squished into shorter and shorter segments.
However, using the log scale is clearly the right thing to do -- or at least try -- for "small" $k$. The OP's graph for $n=10$, as far as he's taken it, looks a lot (but only very roughly) like an amplified sine function, i.e., something for the form $Ax\sin x$ (with $x\propto\log k$), with an increasing amount of noise being added. So it would be of interest to plot the corresponding graphs for other values of $n$, to see if they all have that kind of look to them. In particular, the OP might want to cut back to $n=5$ or $6$, where it should be easy to calculate an entire period.
Actually, because $f(-k)=-f(k)$, the essential behavior of $f$ is captured in the first half of its period, $0\le k\le N/2$. (Note, $N$ is even once $n\gt1$.) So I'd be interested to see the function $f$ plotted using a log scale on $1\le k\le N/2$ for the first few values of $n$. In particular, does the suggestion of an amplified sine wave persist, and if so, can you count the number of cycles it goes through, or does it get washed out by noise by the time you get near $k=N/2$?
It might well be that the log scale isn't quite the right thing to use. (It certainly has trouble near $k=0$.) Someone else may have a better suggestion that offers more insight.
Added later: The OP has added a graph for $n=6$ that suggests that "noise" is going to be a major consideration for largish $k$ even in the half period. (I was going to call it a major "problem" but figure it might be a feature instead of a bug -- in any event, it's not going to go away.) It also finally dawned on me that the function $f(k)$ is perfectly linear for $0\lt k\lt p_{n+1}$, so that first portion looks like an exponential curve when plotted on a log scale, and that portion gets longer and longer as you increase $n$. However, if you normalize things so that the graph for $1\le k\le N/2$ has constant width, that portion will shrink to nothing.