Why toss around "at risk" Soldiers from one unit to several others?

The problem I have found is not limited to suicidal Soldiers but to all “problem” Soldiers. For example, I have heard a 1SG say I don’t want PFC XXX in my Company because he will bring my PT average down. Most of the time I have seen that the “first line” is a SGT that has been in the Army for a few years and got promoted fast. Now before I incur wrath I was a very young E-5 when I made it the only thing I have to thank is some very patient SNCOs that mentored me and helped me along the way.
More towards your point is to get the suicidal Soldier some help more often than not those that may seem like a problem need help and we are far too fast to just Chapter them then address the problem. That “problem” Soldier that you help may one day be a the Soldier of the Year or go even further because someone actually stepped in a helped them.

Just my $0.02

This is a very complex issue. When you are dealing with Soldiers who are "suicidal" you must do whatever possible to mitigate that risk. That is one of our most fundamental responsibilities as leaders. Mitigate risk. Often times, the environment that an individual is in can greatly contribute to the stressors that may be triggering this behavior. It is certainly a case by case basis and it is not always the right answer. But it certainly is an answer for some. I'd much rather try everything within my power to help a Soldier in this condition than have to deal with a suicide and be left wondering if there was something else I could have done to help. If it will help the Soldier, then why not?
I don't look at it as tossing a Soldier around, I look at it as giving them an opportunity to get better in what could be better surrounding for that individual.
This is very different that just moving a problem Soldier around with basic performance issues that you as a leader could not correct.

they just want to get rid of soldiers, easier to send to another unit and that will become their problem. My experience, I got my suicidal soldiers help personally to include sending them to treatment at 28 day non Army inpatient programs.