To dig through more than one layer of bricks, start by digging as many bricks as there are layers, jump in, dig one brick less and so on.
If you dig next to a ladder or a succession of bars, you can dig down through any number of layers and up to five or six bricks horizontally. Just return to the ladder or bars after digging each layer, then move down one square, run out and dig the next layer.
If a digging puzzle looks impossible, consider that there might be some false bricks in it. Maybe the bottom layer is false or maybe you can stand in a false brick and dig next to it.
Think about the order in which you will collect the gold. If you pick the right order, the digging might be easier.
Some levels require you to dig away a brick wall, run in, collect some gold and run out again before the bricks close up and trap you.
It is possible to dig while falling through empty space, if you can get the timing right. You do not need to have an enemy to ride upon. This is a bug which has become a “feature” and is used extensively in the “State of Terror”, “Count” and “Curse of the Mummy” games.