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.