Suppose I have a set $X$ with partition $X_1, \ldots, X_k$. Suppose $Z_i = (z_{i1}, \ldots, z_{iN/k})$ is a random sample of $N/k$ elements from $X_i$ and $Z$ is a random sample of $N$ elements from $X$.
Can I treat "union" $\cup_i Z_i := (z_{11}, \ldots, z_{1N/k}, \ldots, z_{k1}, \ldots, z_{kN/k})$ of random samples from $X_i$ as a random sample of $k \cdot N/k = N$ elements from $X = \cup_i X_i$?
The process you describe is not equivalent to random sampling of N elements from $X$. Suppose $N$=100 and $k$=5, which means each $Z_i$ has 20 elements. Suppose further each $X_i$ has 40 elements. If the random sample chose elements 1 through 19 from $X_1$ then it can only choose one of the rest of the elements (elements 20 through 40). Which means that your process introduces dependencies which aren't there in the straightforward sampling procedure.
The dependencies can be reduced (but not completely eliminated) by making sure the ratio $\frac{N/k}{|X_i|}$ is small for every $i$.
