RAM and data copying/moving
This has been done, mainly in graphics context.

One early variant was the Blitter chips used in several video games and computers in the 80ies, which allows copying and simple processing without involving the CPU. These chips also had masking operations that allowed leaving out some elements from the operation.

The drawback is that since these share access to the RAM, this would change timing while the operation was running, slowing down execution on the main CPU and making execution times unpredictable, so timing sensitive code would stop working.

Any graphics card with dedicated memory essentially behaves like special RAM with included blitter operations from the point of view of the CPU. Several vendors, most famously Tseng Labs (later bought by ATI) also made special RAM (SGRAM) that could run simple operations like XOR (used for drawing window borders during drag operations) or fill blocks in a single operation.

For system RAM, having a predictable access pattern is more important for good system performance than offloading work, so it is actually more efficient to perform copies from the CPU, where they can be scheduled and merged together with other accesses in a single RAM controller, than in a separate unit.