  We investigate low-mass central galaxies with Mstar = $10^{9.5}-10^{10}$
Msun/h, located near massive groups and galaxy clusters using the TNG300 and
MDPL2-SAG simulations. We set out to study their evolution, aiming to find
hints about the large-scale conformity signal they produce. We also use a
control sample of low-mass central galaxies located far away from massive
structures. For both samples, we find a sub-population of galaxies that were
accreted by another halo in the past but are now considered central galaxies;
we refer to these objects as former satellites. The fraction of former
satellites is higher for quenched central galaxies near massive systems: 45% in
TNG300 and 17% in MDPL2-SAG. Our results in TNG300 show that former satellites
were typically hosted by massive dark matter halos (M200 $\geq 10^{13}$ Msun/h)
at z$\sim$0.3, followed by a drop in halo mass at lower redshifts. In addition,
we find a strong drop in the total gas mass at z$\leq$1 for quenched central
galaxies near galaxy groups and clusters produced by these former satellites as
well. By removing former satellites, the evolution of quenched central galaxies
is fairly similar to those of the quenched control galaxies, showing small
differences at low-z. For MDPL2-SAG, former satellites were hosted by less
massive halos, with a mean halo mass around $10^{11}$ Msun/h, and the evolution
remains equal before and after removing former satellites. We also measure the
two-halo conformity, i.e., the correlation in the specific SFR between low-mass
central galaxies and their neighbors at Mpc scales, and how former satellites
contribute to the signal at z=0, 0.3, and 1. The conformity signal decreases
from z=0 to z=1 in MDPL2-SAG but it increases in TNG300. However, after
removing former satellites in TNG300, the signal is strongly reduced but almost
does not change at z$\leq$0.3, and it disappears at z=1 (abridged).
