A block travelling in the air breaks into two parts, and these...

When a block is travelling in the air, it is continuously acted upon by the external force of gravity (and potentially air resistance).

According to Newton's second law, the rate of change of momentum is equal to the net external force. Since the net external force on the system is non-zero, the total momentum of the system is not conserved and must change over time.

When the block breaks into two parts and they separate, the internal forces of the system do positive work. This process converts internal energy (such as chemical or elastic potential energy) into kinetic energy.

The total kinetic energy of the system can be written as the sum of the kinetic energy of the center of mass and the kinetic energy of the parts relative to the center of mass ($K = K_{cm} + K_{rel}$). Since the parts separate, they acquire relative velocity, causing the relative kinetic energy $K_{rel}$ to increase from zero to a positive value. Thus, the total kinetic energy of the system must change.

Therefore, both the total momentum and the total kinetic energy of the system must change.