Writing about about some brand new cars, I would write

des voitures flambant neuves

Larousse states that:
http://www.larousse.fr/dictionnaires/francais/flambant/33977/locution?q=flambant#172263

Flambant neuf, entièrement neuf et brillant (expression le plus
souvent invariable)

But also that:
http://www.larousse.fr/dictionnaires/francais/flambant_flambante/33977/difficulte

Flambant reste invariable dans l’expression flambant neuf : une voiture flambant neuve.

So, in the “flambant neuf” expression, the adjective “flambant” is always invariable. But for the adjective “neuf”, both variable and invariable versions are valid.

You have the gist of it, that is whether in the phrase flambant neuf, flambant is considered as a present participle (no agreement) or as an adjective (agreement).

Le bon usage (Grevisse, 10th edition, 772,B):

Dans les expressions familières (tout) battant neuf, (tout flambant neuf), la forme en -ant reste le plus souvent invariable.

Grevisse gives numerous examples with no agreement on either word and the agreement on neuf only is more frequent than the agreement on one both words.

Des habits flambants neufs (Alexandre Dumas, Le Comte de Monte-Cristo)

Des soldats battants neufs (Maurice Barès, Colette Baudoche).

He only gives one example with the gender agreement :

C’est parce que la Belle-France est flambante neuve (G. Chèrau Champi-Tortu)

and he adds that the gender agreement is rare.

The TLF makes a note about the fact that flambant is now considered as invariable but neuf can agree it is considered an adjective or an adverb:

Dupré 1972 indique l’invariabilité de flambant semble acquise dans cette expression. Mais on observera (…) l’indécision de l’usage sur l’accord de neuf, traité tantôt comme adjectif tantôt comme adverbe`

• Deux grandes valises semblables, flambant neuves, en peau de porc (Butor, Modif.,1957, p. 11)
• Librairie flambant neuf, tout ensemble radieuse et dégoûtante d’argent récent (Du Bos, Journal,1927, p. 190)

The BDLQ says that flambant does not agree but that neuf generally agrees. It’s what is generally taught at school, e.g.:

L’ajectif flambant dans la locution flambant neuf est toujours invariable, mais neuf peut varier en genre et en nombre ou rester invariable : une voiture flambant neuf ou une voiture flambant neuve.

Usage, tends to reflect the agreement on neuf and and not on flambant, although we can find recent examples such as:

Vestiaires, buvette, flambants neufs à La Noyère (29/02/2016)
Des locaux flambants neufs (28/06/2017)

An ngram reflects what Grevisse says, i.e., that the agreement on gender is rare. flambants neufs is much more frequent than flambantes neuves:

The Parlons Français blog’s advice is to say:

Des gradins flambant neuf ou, plus couramment, flambant neufs (de préférence à flambants neufs).

It is quite a good blog but it not an authority.
The Académie française does not (yet) have an entry about flambant neuf. I do not have a Bescherelle at hand but it would be worth looking into it.