This subject to some is a bit touchy, mainly because "colour temperature" is an evasive property, and per se, does not exist in any gas discharge lamp, as electronic flash.
The colour temperatures of daylight and flash are "perceived" colour temperature, but do not inherently cling to the K scale, as they are not emitted from a black body.
The black body radiation defines the K scale, as a black body will radiate that colour temperature when heated to the same Kelvin temperature. However, a characteristic of black body radiation is a richness in red and infrared, while the blue and violet as well as ultra-violet are not so pronounced. Between those extremes, black body radiation has a smooth spectral curve, with no spikes or notches, falling evenly toward the higher frequencies.
Sunlight is not emitted from a black body. Any star, Sun as well, is a large nuclear reactor, mainly combining hydrogen into helium. When the very jagged spectrum of the sunlight hits the atmosphere, it is largely dispersed, and much of its blue radiation is spread over the entire globe, while the longer rays hit the surface of Earth more directly. The daylight spectral curve does not have the smoothness of the black body radiation. Neither has the studio flash.
In essense, this means, that if the studio flash has a spectral curve similar to that of daylight, it is possible to get similar tonal reproduction using a flash as with daylight. However with true K radiators, as incandescent bulbs, it is impossible to get similar colour rendition. It can be accomplished with a parametric filter to change the spectral curve, as neodymium lamps do, even though those lamps have a rather low colour rendition index. The metameric response may then be similar to daylight, when white balancing for the lower K value of the lamp.
So without filtering, the metameric response from daylight and K radiators (incandescent lamps) cannot be similar, are not equal; never.
The lighting industry has coped with this problem by adopting two different K scales, one for black bodies, up to 4999 K, and another from 5000 K and above. The one for values from 5000 and above is called "daylight".
And of course an electronic flash, by design, is consistent. At least as long as its energy is consistent, the discharge from the capacitors.
We might appreciate, that skin tones of a person is never seen in the light of an electronic flash. The electronic flash has its own properties, and the metameric response will be different from natural daylight, although it may be consistent. It is however different from any way that we may see skin tones naturally.
It boils down to the simple fact that there is no "dead on" perceived skin tone, even if there may be a "dead on" white balance, as measured from a white reference. The white reference or measurement directly from the light source, can adjust our three channel processing of the RAW data to the colour of the light source, but the metameric response, the tri-stimuli response in our eyes, the skin tones, will depend on how the light is reflected from that particular skin. It will be different, with any different light source.