Some things to be aware of:

Your concrete probably has control joints (deep groves that provide a discrete weak place for a crack to happen) or expansion joints (a complete separation between slabs). When you put flagging over these you either have to break your patterns to allow the joint to function or risk having damage to your finished surface if there is either freeze/thaw or settling cracks (more often than not).

Another consideration is that you will be adding at least an inch to the surface of the patio which will throw off any riser height to existing steps which can be a major hazard. If your steps are concrete, you can add flagging to them as well, but it will make the top step higher than it is now - sometimes this is fine if there is a landing and you have enough of a gap between the threshold and the landing to take on the stone. You will have to be very careful on how the flashing is done.

What Laag said, plus:

Flagstone isn't easier to sweep than concrete, and will also get stained. The inevitable cracks or dips in your mortar and stone will collect bits of leaf which will decompose and provide a place for weeds to sprout. This would even happen with tile (stone or ceramic), which might be a thinner alternative material to consider.

You could just have the concrete stained, with or without a design etched in, to get a surface that will show the leaf staining less. Or, just wait till the staining changes the overall colour of the concrete naturally.

KarinL

I used Indian slate tile over concrete patio so I could 'honor' the expansion joints. Looks fabulous. And very easy to keep looking great by blowing off leaves/grass clippings.