Lots of bombers were used for logistics in the first few months after the war. But the US had by far the largest number of real transports and with the drastic reduction in the size of of the Armed forces in late 1945 and 1946 the need for using unconverted or minimally converted bombers as transports disappeared real quick. By Dec of 1947 (granted 2 years late) the War Assets Administration was announcing that remaining Curtiss C-46 transports (about 627 aircraft) would be sold for $5000 each! If they didn't need C-46, C-47 and C-54s in 1947 they sure didn't need the old bombers. They had built over 3100 C-46s, how many had crashed or been grounded by Sept 1945 I don't know or how many were kept as not being surplus to requirements after 1947 I don't know.
As far as using the the B-24s as combat aircraft goes.
From Wiki: "While the end of World War II caused production of the B-29 to be phased out, with the last example completed by Boeing's Renton factory on 28 May 1946, and with many aircraft sent for storage and ultimately scrapping as surplus to requirements". If you have more B-29s than you can man and use with the much reduced peacetime establishment why on earth would keep B-24s in service?