Why can I not punch or break blocks while moving in Minecraft?

You need to update your graphics or find a new server, try to use this:
/register minecrafter minecrafter

If you are playing survival mode you will have to hit a block several times to break it.

Also there's another gamemode called "adventure" which is terrible… You can only break blocks with the intended tool. However, if you can break them while standing still I don't think this is the case.

Try typing /gamemode 0, /gamemode 1, or /gamemode 2 to switch your game mode to Creative and see if you can break them while moving then… If you can, then it's just that you need an upgraded tool, etc.