Alternate picking across strings can be quite challenging. If you think about your pick motion as you move from a downstroke on string 6 to 5, your pick is moving in a downward motion but you actually have to miss string 5 and come back up on it with an upstroke if you're following pure alternating picking. This makes for some tricky picking when you first start it. Once you are comfortable with alternate picking, cross string picking shouldn't take much effort.
All of the examples are using the A minor pentatonic scale except for the last open string exercise and are a continuation of the downstroke cross string picking exercises.
This exercise focuses on moving from the low to high string by picking the next adjacent string to the one you are currently on. Really watch your picking hand to make sure it is picking correctly.
This exercise gets harder because now you are jumping over one string as you move from low to high.
Jump 2 strings at a time. Move from low to high and then come back down.
Just like the downstroke cross string picking exercise with the open strings, this exercise is pretty much the same but you need to alternate pick. Again the hardest part of this exercise is to mute or dampen any open strings as you move off them. Start by playing this exercise extremely slow.
A classic cross string picking song is Sweet Child O' Mine by Guns N' Roses. The main intro riff contains a lot of cross string picking and to play it up to speed you will need to utilize alternate picking.