IS Array of Positive Integers Always 'Equalizable' With This Strategy?
If the GCD of the array D is larger than 1, then this cannot always be done. The simplest example is $A = \{1,2\},D=\{2\}$. In contrast, if the GCD of the array D is 1, then it is known that large enough integers can be represented as positive linear combinations of elements of D. In particular, there is an integer $N$ such that $N-a$ can be represented for all $a \in A$. Summing all these representations will equalize the array.

For example, when $A = \{2,3,7\}$ and $D = \{1,2,5\}$, we can add $\{1,0,0\}$ once and $\{1,1,0\}$ four times, reflecting the representations $7-2=5 \times 1$, $7-3=4 \times 1$, $7-7 = 0 \times 1$. Even when $D = \{2,5\}$ this has a solution: using $11 = 3 \times 2 + 1 \times 5$, $10 = 2 \times 5$, $6 = 3 \times 2$, we can add $\{2,0,2\}$ three times, $\{5,5,0\}$ once, and $\{0,5,0\}$ once to get $\{13,13,13\}$.