Neutrinos are produced in beta decays with a given helicity. My question is wether this helicity is a constant of this movement or is it variable?
Your starting point
Neutrinos are produced in beta decays with a given helicity.
is plain unsound. Neutrinos are produced with a given chirality, which, for massive neutrinos, is distinctly different than helicity, \https://en.wikipedia.org/wiki/Chirality_(physics)#Chirality_and_helicity.
Consider a positron emission β-decay, so, effectively, the disintegration of a virtual $W^+$. A right-chiral positron is produced in the nucleus' frame, in association with a left-chiral neutrino (actually a massive "electron neutrino", which is a "fictitious" convenience linear superposition of three massive neutrinos, but let's not muddy the waters, for now).
This left-chiral neutrino will have a wave-function with a huge component of negative helicity, and a minuscule, $O(m_\nu/p)$ component of positive helicity, normally ignored, as it would vanish for a massless neutrino and underlies your confusion about "definite helicity production".
As the comments indicate, this mix of helicity components is frame-dependent, as always. So, if you scooted after the neutrino and you overtook it, it would appear to be receding from you in the opposite direction: so it would now be mostly of positive helicity.
(There is, of course, no such thing as left- or right- helicity... terms virtually invented to confuse. Use at your own risk.)
Frame dependent or not, helicity is a wonderfully useful concept, much more intuitive than chirality, and has simple accounting rules for physical systems, which is why it is routinely used.
For more formal detail, you may see this.