The simple frame of mind you should have when letting people go or leaving yourself is that it's a small world and you might end up working together, interacting, meeting at a conference, etc. in a different context at some point in the future. So, don't burn any bridges.
An employment relation is a business relation ship with some contractual obligations and rights. It's all spelled out in the employment contract. Anything on top of that is the personal integrity of the employee and the reputation of the company. It's a mutual interest to do the right thing on both sides and behave in a way that is consistent with maintaining a good reputation.
For employees: your actions at company A might cause you problems at company B. People talk. I've seen people torpedo their careers by basically getting a well deserved reputation for not being entirely reliable/trustworthy. Anger issues in particular can be a problem. If you have them, learn to deal with them. I've actually advised friends to not hire certain people based on my negative experience with them. I've also done the opposite and helped people out by introducing them to their future employer. My own career changes each involved people in my network recommending me or introducing me. So, don't burn any bridges even if your employer treats you badly. Keep it friendly and professional.
For employers: people talk about their employers both inside and outside the company. If your reputation suffers because of how you treat your own employees, it becomes a problem for recruiting. The more disgruntled ex-employees you have, the harder it becomes to recover from that. You can offer more money to compensate but your actions will cost you. Worse, your own employees might start voting with their feet if they see you treat people badly. It's usually your best people that leave first when that happens. Because they have options and good reputations. If the A's leave, you end up with a lot of B's hiring C's.