We know the Earth's interior is differentially rotating -- the core is rotating much more quickly than the crust. This is possible because the outer core is liquid and is presumably low enough viscosity that it doesn't transfer angular momentum outward very quickly.
Some other planets/moons have solid (or near-solid) interiors that rotate at the same rate throughout.
What would happen as a planet -- let's say Mars -- cooled enough that the outer core solidified and (like a gigantic planetary clutch) connected the inner core angularly to the mantle/crust? Would it happen gradually enough that it would unnoticeable to any surface dwellers? (other than eventual extinction due to cosmic radiation as the magnetic dynamo faded)
My question is inspired by a recent article about Mars' ancient dynamo possibly still recorded in surface ferromagnetism. http://m.phys.org/news/2016-11-mars-ionosphere-crustal-magnetic-fields.html
Update: as JohnRennie points out below, this question is based on a faulty assumption that the Earth's core rotates much faster than the crust. In fact it takes about 400 years(!) for the core to rotate one more time than the outer portions of the planet. Instead, the dynamo is caused by convection cells in the outer core.