We show that the range of a critical branching random walk conditioned to survive forever and the Minkowski sum of two independent simple random walk ranges are intersectionequivalent in any dimension d ≥ 5, in the sense that they hit any finite set with comparable probability, as their common starting point is sufficiently far away from the set to be hit. Furthermore, we extend a discrete version of Kesten, Spitzer and Whitman's result on the law of large numbers for the volume of a Wiener sausage. Here, the sausage is made of the Minkowski sum of N independent simple random walk ranges in Z d , with d > 2N , and of a finite set A ⊂ Z d. When properly normalised the volume of the sausage converges to a quantity equivalent to the capacity of A with respect to the kernel K(x, y) = (1 + x − y) 2N −d. As a consequence, we establish a new relation between capacity and branching capacity.