Heat Pumps and Electric Heating - Heating under kitchen cabinets

I have a new kitchen over a crawl space. The plumbing pipes, which run along base of the exterior wall (but inside the house), tend to freeze up at very low outside temperatures because the radiant heat in the floor does not extend under the cabinets.
Should I add heat somehow through the toekicks or should I do the reverse - blow the cold air out from the toekick thereby bringing the ambient air in through vents in the toekick? Any other ideas?

You could also try wrapping it with heat tape.

I don't quite understand where the pipes are located. If they are in the wall or below the floor in the crawl space. If they are under the crawlspace, you can insulate them or use a heat 'tape' that you wrap spirally around them that will keep them above freezing, or both.
If they are in the wall, behind the wall surface, you need to improve your insulation or decrease the air infiltration, or both. If they are running inside the house, underneath the cabinet toe board, then you could just open up the area to natural circualtion with a grill or multiple placed along the face or end of the cabinets.
Hope that gives you some ideas.

By the way, there is a big question going on about whether crawl spaces should have the floor sealed up with plastic sheeting, and closed up from external air flow. Some recommend using thick plastic, and preventing below ground moisture from getting in and then sealing from outside air to form a pocket of more or less still air that is not humidified by the ground or outside air. I don't have a horse in that race...
tom

Whether crawl space part of building envelope or not, the vapor retarder is needed to retard vapor emissions from soil into crawl space area. The vapor retarder is overlapped, taped, and run up sides of foundation where it is attached with silicone caulk. Closing off foundation vents and including the crawl space as part of the building envelope is a good option as long as crawl space stays dry and there are no moisture issues. This can be discussed with heating/cooling expert.

As indicated, if pipes are accessible, they should be insulated. During periods of deep freeze, keeping kitchen sink base doors open for better air and heat circulation is helpful.

It is not clear if plumbing pipes "which run along base of exterior wall" are in crawl space or in the wall behind the cabinets or "in the house" inside the kitchen and running behind or running through the cabinets in toe kick area.

Location will determine what you can do to the pipes, such as insulate or put vents in toe kick. It is not clear whether or not your walls are insulated. If not, blown in insulation is an option.

Many keep a pencil lead size stream of water running on nights during deep freeze to keep water from freezing and keep sink and vanity base cabinets open for better heat circulation. Insulation of pipes is crucial. Sealing around where pipes enter sink and vanity cabinets will seal cold air out of cabinets.

Some posters report stacking bales of straw or hay along exterior walls where pipes are located to add extra insulation and prevent freezing. With a little more investigation to assess your situation, it is possible to prevent frozen pipes.

Water pipes are inside the house underneath the cabinets at the rear of the toekick, i..e. they are hard to reach. The space is new construction (believe it or not) with insulated walls and R40 including foam board under the flooring. The 'crawl' space has a concrete floor, 8 feet tall and fully enclosed to outside except for a flood vent which is required. Ideally, I would just heat the underneath space, but the local building inspector has required a flood vent even though the area always has been part of the house footprint. The problem is that little heat reaches the space under the cabinets because the plumber would not put even minimal radiant tuning under the new cabinets.

From your sensible comments, I sense that I should try vents in the toekicks first. My instinct tells me, though, that vents alone will not provide enough circulation of warm air.