Although Shakespeare's history plays deal with the morality of actions, I wouldn't say that they are based in the English morality plays. Shakespeare's source for his history plays is Holinshed. For his history cycle from Richard II through Richard III, he was dealing with fairly recent history. Elizabeth I was a Tutor, granddaughter of Henry VII who defeated Richard III at Bothsworth Field which ended the War of the Roses.
It must be remembered that Shakespeare was a dramatist not a historian so he took license with history. If Richard III was not the monster created by Shakespeare, it was politically safer. Whether or not the man created by Shakespeare is indeed the same as the king, we really don't know. Tutor propaganda spun the truth to suit their needs. We don't know if Shakespeare knew the truth or not but for a dramatist it was wonderful material
Throughout the series of eight plays (Richard II, Henry IV 1, Henry IV 2, Henry V, HenryVI 1, Hnery VI 2 Henry VI 3, and Richard III) Shakespeare explores his own and England's recent past.
"Uneasy lies the head that wears the crown" is revealed in various ways. Richard II discovers that when he sends his cousin, Henry into exile, then takes his family lands. Henry usurps the crown and finds himself having to continually fight to keep it. His son proves to be a valiant leader who dies young in the Hundred Years War. His son would have made a better priest than king and not only has to fight the Hundred Years War but also a civil war aka The War of the Roses with families torn apart and himself murdered. Finally Richard, the master villain.
Through this all, Shakepseare was able to create wonderful and human characters all nurtured by warfare.
There are several important lessons here. Shakespeare reflects warfare from the heroic to the cowardly but more importantly the horror of warfare. Warfare seems to bring out the worst in mankind. Families are literally torn apart and entire societies suffer as a result of the getting of and keeping of the golden round.
Richard III is the culmination of it all. Richard even bitterly reflects on this in his opening soliloquy. Whereas the plays that precede Richard III are sweeping landscapes, battlefields in France and England, this play brings the whole family argument in to a microcosmic focus. We listen to Richard tell us how he will get what he wants and we then watch as he does exactly what he says. He gets the throne but has trouble keeping it. In the end, the war is over and the Tutors bring peace and stability, relatively speaking.
Another lesson is child kings. We see the ineptitude of both Richard II and Henry VI, both child kings who never learned how to rule. In Richard III, we have the possibility of another child king. If Richard did kill the little princes, it would appear that he did England a favor since the Woodville family would have had the power. This would not have been a good thing.
Since in the end of Shakespeare's history plays, it could be said that good conquers evil, it could be said to be a morality play, but it is much more complicated than that.
I moved your reference to Richard III to a more prominent place in your question, since I think it is very key to answering your question.
First, you should know that as an English playwright, it would be nearly impossible for Shakespeare NOT to use the Miracle, Mystery and Morality plays and how they operated on an audience as the foundation upon which he developed his work. He was a man of the theatre first, and this would have been the street theatre that he grew up on.
That said, this is a broad question, so I'll point to a couple of pertinent connections between Morality plays and Richard III, but you could certainly investigate further for much more richness of detail regarding this comparison.
Morality plays were meant as teaching tools presented as performance by the Church to a largely illiterate audience. They were meant to show ordinary men (Everyman is the most famous) being confronted with choices between the good, Christian road and the evil path of Satan.
As relates directly to Richard III, two things are very significant. First, the actors in these plays not only attempted to convince the "Everyman" of making the good or evil choice that they personified, but they also worked to convince the audience. Therefore, they spoke directly to the audience. The audience was never a passive observer but rather a co-protagonist. This is important, because from the opening words spoken by Richard, he is directly related to the Vice character and, as such, attempts to convince the audience of the validity of his course of action. He addresses and challenges them directly.
Which brings us to the second connection I'd like to make to Morality plays. Richard III is often criticized for a protagonist that isn't "three-dimensional." I would suggest that this was an intentional choice that Shakespeare made, as he directly drew upon the one dimensionality of the characters in Morality plays, characters representing qualities like Mercy, Charity, Lust, Sloth, etc.
Certainly, there is much more to unearth on this topic, so good luck developing this further!