If you have an old pair of boots on which the leather of the boot shafts has become stiff, it is likely they are so stiff because the leather has dried out. To remedy this problem, buy a good quality leather conditioner, such as Lexol leather conditioner (which higher-quality shoe or leather stores may stock, or you can find it on-line). Apply a small amount of it onto a lintless cloth, and rub it gently in circles on the leather. Do that all over both boots – the shafts and the feet (but not the soles.)
Leather conditioner can help replenish the oils and moisture naturally found in leather and make the leather more soft.
By all means, do NOT use paste wax, like Kiwi. Wax clogs up the pores in the leather and actually causes it to dry out more quickly. If you have wax build-up on your boots, remove it by gently wiping a solution of diluted white vinegar (1/2 cup vinegar to 1/2 cup water) with a cloth onto the boots. That should remove the wax without damaging the leather. Let the boots dry out completely after removing wax, then apply leather conditioner as described above.
Leather conditioner should make the leather more flexible. Depending on how old and how dry the leather is, you may not be able to make the leather as soft as it was when the boots were newer.
Do not use water to soften the leather. Initially, water will make stiff leather more flexible. But when the water evaporates, the thin bonds that hold the cellulose fibers that compose leather will break. Evaporation and drying out of leather cause by soaking with water will cause permanent damage. Leather conditioner used sparingly is the best way to restore natural oils and flexibility to leather.