Lime has caused major buildings to look streaked for months and even years. On large projects, designers will specify testing and quality standards for lime content, leaching, and cleaning methods.
On your front stoop, you probably did not have a large firm planning and checking to see if the mix was controlled. The leaching of lime, referred to as efflorescence, can continue for a very long time.
What causes leaching is water that has no other path but to be drawn into the brick or stone mortar joint, then drawn back out again by changes in temperature and air pressure. The way to prevent or limit the problem is by making an air space behind the face brick, placing holes at the bottom of the wall, or stoop, and allowing the masonry to drain. In this way, the moisture has a better path, so that it will not be inclined to carry lime deposits from the inside to the outside surfaces.
You basically have to bypass the problem. In your case, I can safely bet that water gets trapped and then is forced out, taking some of its lime buddies to freedom. Power washing can accelerate the leaching problem at the surface, and may have to be done several times with a mild detergent.
Be very careful with sealants, such as the water seal applications you see advertised. If the surface is completely covered, you can smother the brick, causing the surface to crack and fall off.
Cleaning is the best solution.