Mars-crossing asteroids (MCAs) constitute a population of transient bodies coming from the asteroid belt that end up in the near-Earth region or are ejected hyperbolically. We analyze the dynamical evolution of a sample of 1066 MCAs with absolute magnitudes H<16 for 800 Myr. We find that the sample can be split in two groups: the high-inclination group (i≳15°) and the low-inclination group (i≲15°). Asteroids of the first group generally have much longer dynamical lifetimes than those of the latter group. We focus our attention on the largest members of the MCAs in high-i orbits which seem to be in excess relative to those of smaller sizes. We followed in particular the dynamical evolution of (132) Aethra – the largest MCA – and 9 clones for an extended period of 2 Gyr. We found that Aethra and all its clones remain in a stable orbit for the whole studied period. Despite Aethra partially crosses Mars’s orbit, no close encounters with this planet are registered (within a Hill radius) which is due to the Kozai mechanism by which the oscillations of Aethra’s perihelion distance are coupled with the precession of the argument of perihelion. Given its size, the Yarkovsky mechanism, that causes a size-dependent drift in the semimajor axis, is probably too small to drive Aethra’s semimajor axis to an unstable zone in the asteroid belt on a Gyr timescale. Long dynamical lifetimes (at least several hundreds Myr) are also found for other large MCAs (D≳10 km) in high-i orbits, which also share Aethra-like dynamics. Their long dynamical lifetimes may explain their observed excess in comparison with what would be expected from the extrapolation of the size distribution of smaller members of the population, whose dynamics may be affected by the Yarkovsky mechanism that destabilizes their orbits and shortens their lifetimes. By contrast, there are very few large MCAs in the low-i population, which suggests an insufficient replenishment rate of big bodies in low-i orbits. The time scale to shift big fragments produced in catastrophic collisions to unstable zones of the main asteroid belt by the Yarkovsky mechanism may very likely be too long in comparison with the dynamical lifetimes of low-i MCAs. The frequency of collisions of the Earth with any of the large MCAs is estimated to be of one every several hundreds Myr and it will most likely move on a low-i orbit.