The trick is to use a starter pin punch to get pins moving. Once they're moving switch to a longer punch to fully remove.
A starter punch is just a standard punch cut off to about 1/2 inch working length. The shorter length makes for a much stiffer punch that won't flex or bend.
In many cases you can use a nail set, which virtually can't flex or bend to start pins moving.
As for punches, I used the Brownell's replaceable pin punches.
They sell all sorts of lengths from starter pins, to roll pin, to punches with concave ends for round head pins, and others.
They come in at least three sizes and many lengths.
The best part is, bend or break a pin and instead of waiting for another to get shipped to you, you can simply unscrew the collar and install a new pin.
Also, with three handles you can use all the sizes, lengths, and types of pins.
It's also much cheaper to buy a few spare pins then having to toss the entire punch and buy a new one.