Hi Lex,
There are a number of ways to approach a board depending upon your riding skill level and also the way the board is positioned ie upside down or right side up.
If it has a handle and is right side up, I approach at a moderate speed and grab the handle whilst steering and keeping the bar in for full power. This will be easy on a kite with low bar pressure and easy responive turning. You dont even have to lift the board out of the water and in fact can ride your board and push the other board along the water (planing) with your hand on the handle.
Alternatively, you can stop and plop in next to the board and then place the board across your lap in a water start position. This takes some practice when beginning but again you need to be able to sheet in, hold power and steer the kite into and out of a power stroke with one hand.
Be careful grabbing boards that are upside down as you can easily cut your hands on sharp fins (personal experience), like the old North boards fitted with the TFC fins which have the sharpest trailing edges around.

Good winds,