Generally speaking, you cannot use memory in a system that is slower than the memory that the system is designed for. If you put faster memory into a slow memory slot, the memory is supposed to run a the slower speed (I have had mixed success with this), but putting slower memory into a fast memory slot will not work because the memory will not be able to get the information to the CPU as fast as it is expected.
In your particular case, the "old" machine uses DDR3-1333 memory and the "new" machine uses the faster, lower voltage DDR3L-1600 memory. A page at Dell explains that "DDR3L is a dual voltage capable memory SoDIMM, which supports operation at both 1.5V and 1.35V." So you could use DDR3L modules in places that ask for a DDR3 module, since it can operate at 1.5V, however "DDR3 is a single voltage capable memory SoDIMM, which supports 1.5V operation only."
The macmini5,1 machine uses an Intel Mobile Core i5 "Sandy Bridge" (I5-2415M) CPU and the macmini6,1 uses an Intel Mobile Core i5 "Ivy Bridge" (I5-3210M) processor
The Intel website does indicate that the newer 3210M CPU does have support for both DDR3 and DDR3L, but I cannot tell if Apple's other systems on the MacMini motherboard would support it.
A component designed for a higher voltage seems unlikely to cause any electrical damage if put into a lower voltage system (it will draw lower current at the lower voltage, which seems unlikely to cause any damage), but running it at the "wrong" voltagae, and at a higher speed than it is designed for, seems very unlikely to be workable.
If you do try it out, let us know how it works.