>>>M-up and M-down also move paragraphs and most blocks (in Org 7.9).
>>>Unfortunately, binding is shadowed when trying to move a src block.
>>
>> When I try that I get these messages:
>> Cannot drag this element forward. 
>> Cannot drag this element backward.
>
>Can you share the file on which you tested this?

Here is the file. Most elements have been removed. The test was under today's date. 

>If there is only one paragraph under a headline, you cannot move
>it up and down, the move is constrained by the structure of the
>file.
>
>Check also C-M-t (Control Meta t) to transpose elements and
>S-M-<up/down> to move a line up and down (this last command is
>structure-agnostic and can be handy sometimes.)

Hmmm.... 
S-M-<up/down> should be the ticket but doesn't work for me. 
Assume this is SHIFT-ALT (held down together) up arrow. 
Also tried SHIFT-ESC up.