Is there any shortcut key for moving a component by its name (designator)?
In CS this can be done from the menu "Tools --> Move --> Component --> mouse left-click" on the working area. In the pop-up window that arises, you can type (or select from the dropdown list) the name of the component you want.
You can select as many components as you wish if you use continuously this tool.
However, if you "escape" from this tool e.g. for "routing" some tracks, you have to repeat the above time-consuming process.
For example, in my other PCB software, you can simply press "F4" each time you want to select --> move a component on the layout, assuming that you follow the corresponding schematic document.