Yes, there is a speed of motion and the front end of an object will start to move before the back end does.

So let's look at your thought experiment. Imagine an extremely long metal rod in space. We know metal is highly resistant to compression and not very elastic. If you apply too much force at the front, the rod will deform. If you apply a force that is sufficiently low such that it avoids permanent deformation and that force is applied over a long enough time, and the rod is sufficiently long enough, yes, you will achieve your 10cm of compression in motion where the front of the rod will have moved 10cm before the back of it will experience any motion.

The length of the rod to achieve this effect would largely depend on the bulk modulus (resistance to compression), strength of the material (you don't want to deform it), and the speed of sound in the material (which is really the speed of movement).

Here is a great video where an experiment is created to answer this exact question. I highly recommend a watch all of the way through.

https://www.youtube.com/watch?v=DqhXsEgLMJ0