According to the french equivalent of the FDA, the ANSM, in a minoxidil presentation Resume des Caracteristiques Produit :
Le mécanisme précis d'action n'est pas connu.
The mechanism of action is not known
According to the following study : Minoxidil: mechanisms of action on hair growth. the mechanism of action is still not known, even if scientist start to suspect some mechanism of action to be the cause of the effects :
There is some evidence that the stimulatory effect of minoxidil on hair growth is also due to the opening of potassium channels by minoxidil sulphate, but this idea has been difficult to prove and to date there has been no clear demonstration that KATP channels are expressed in the hair follicle. [...]stimulation of cell proliferation, inhibition of collagen synthesis, and stimulation of vascular endothelial growth factor and prostaglandin synthesis. Some or all of these effects may be relevant to hair growth, but the application of results obtained in cell culture studies to the complex biology of the hair follicle is uncertain
As you can read, vasodilatation does not seem to be the reason behind the hair growth.
Your hair loss, unless a physician diagnosed something else, is probably normal male pattern hair loss . Studies tend to prove alopecia in male is androgen (testosterone and DHT) related and appear to be caused by a lack, or an inactivation, depending on the source, of stem cells.
It is however not about the lack of nutrition or blood flow of the hair follicle. This is a very (very) vast subject, but the wikipedia "Pattern hair loss" article (can't post more link) provides plenty of valuable information about it.
The warm compress treatment is therefore very unlikely to work and I'd suggest you to ask a physician before attempting anything.