Indexing using the [*, *, sorted_z] creates a list of indices with a 4-byte value for each element, leading to that huge memory usage. In this case, making a second copy (perhaps unavoidable), then looping over one dimension ends up using far less memory and taking less time. (Sometimes a loop is OK, especially when one statement still operates on 4 million elements!):

PRO IndexingTest

nPlanes = 287
sorted_z = Sort(RandomU(42L, nPlanes))

vol = IntArr( 512, 512, nPlanes )
vol[0, 0, *] = IndGen(nPlanes) ; Put [0,1,2...] in position [0,0] on each plane
mem0 = (Memory())[0]           ; Current memory usage
t0 = SysTime(/Seconds)

;;    Method 1:
vol = temporary( vol[ *, *, sorted_z] )
;;

duration1 = SysTime(/Seconds)-t0
mem1 = (Memory())[3]           ; Peak memory usage since last Memory() call
memUsedMB1 = (mem1-mem0)/1024./1024
;;    Test that it actually worked:
Print, 'Test 1 '+(Array_Equal(sorted_z, vol[0, 0, *]) ? 'passed' : 'failed')

vol = IntArr( 512, 512, nPlanes )
vol[0, 0, *] = IndGen(nPlanes) ; Put [0,1,2...] in position [0,0] on each plane
mem0 = (Memory())[0]           ; Current memory usage
t0 = SysTime(/Seconds)

;;    Method 2:
vol2 = IntArr([512, 512, nPlanes], /NoZero)
FOR planeI=0, nPlanes-1 DO vol2[0, 0, planeI] = vol[*, *, sorted_z[planeI]]
;;

duration2 = SysTime(/Seconds)-t0
mem2 = (Memory())[3]           ; Peak memory usage since last Memory() call
memUsedMB2 = (mem2-mem0)/1024./1024
;;    Test that it actually worked:
Print, 'Test 2 '+(Array_Equal(sorted_z, vol2[0, 0, *]) ? 'passed' : 'failed')

Help, duration1, duration2, memUsedMB1, memUsedMB2

END ;; IndexingTest

Output:

IDL> indexingtest
% Compiled module: INDEXINGTEST.
Test 1 passed
Test 2 passed
DURATION1       DOUBLE    =       0.89100003
DURATION2       DOUBLE    =       0.67199993
MEMUSEDMB1      FLOAT     =       430.500
MEMUSEDMB2      FLOAT     =       144.000

Perhaps you could avoid making the copy of the array if you knew what sequence of swapping planes of your array would lead to the desired sorted_z... left as an exercise for the reader :-)

Cheers,
-Dick