Won't war and Global Climate changes generally cause people to move underground?

Yes. This is true to some degree.

We are already seeing this in Ukraine as people head to the basements or subways for shelter from bombs and missiles. Sometimes this works and sometimes they get buried alive too if there is a direct hit on their bomb shelter. However, one is less likely to be hit by shrapnel flying in all directions when bombs or projectiles explode and create carnage over several blocks usually. So, unless you get a direct hit on where you are you are likely more likely to survive whatever happens above you in some sort of underground shelter.

Interestingly enough though, this also applies to storms especially wind storms like hurricanes and tornadoes where if you are sheltering underground or at least under an overpass on a freeway or something that you are more likely to survive a tornado especially this way if you take a direct hit from one or a direct hit from a hurricane.

For example, because it's cold out now still as so many cities in Ukraine are getting blown up, going down underground in Subways and basements keeps them alive except there is no electricity or heat or food or water most places now unless they stocked up before Russia invaded Ukraine.

However, obviously, moving underground won't work in a flood plain where water is going to go down where you will be some or all of the year. In that case maybe you need a boat that sleeps the number of your family for a flooding emergency maybe. 

But, some locations would be ideal for digging underground, for example into the side of hills or mountains, depending upon what kind of dirt or rocks you are digging into. For example, you don't want to dig somewhere where it collapses on your family.

So, it's a good idea to have an building engineer or Architect or building inspector look at where you are planning to dig so he or she can offer suggestions on whether it is safe or not and how expensive it will be to do what you want to there.

However, Yes. Generally, I think people around the world will start to move underground just watching what is happening to cities in Ukraine being leveled to the ground with all power and water and food shut off by an invading army and tanks.