Many polygynous families are patrilineal/patrilocal. Women are in a weak position and may be victimized.
Rivalries between segments (half siblings) tend to be intense.
It is extremely difficult to keep everyone happy (unless there is really a lot of money).
Next: Unusual systems of marriage
Polygynous families appear to have great solidarity from the perspective of people who are are not family members. Family members will tend to unite and defend against any threat from non-members. However, polygynous families also have structures that make certain kinds of solidarity and competition common. Siblings who share the same mother tend to have much greater solidarity with one another than siblings who share the same father but different mothers. In Bambara (the language spoken by many in Mali), this is captured by the ideas of badenya (mother-child-ness) and fadenya (father-child-ness). Badenya stands for home and hearth, warmth and comfort. Fadenya stands for rivalry, conflict, and division. But it's important that badenya also implies stasis and boredom. Fadenya, change and adventure.
One additional thing that needs to be considered: polygyny implies different sexual morality for men and women. In polygynous marriage, women are, in a sense, really married monogamously. That is, they are married to a single man. Not only are romantic or sexual liaisons with other men prohibited, in some societies they can be brutally punished. Men, on the other hand, by definition, are expected to have romantic and sexual liaisons with multiple women. So, there is a different sexual morality for men and women. Defenders of polygyny point out that 1) men who have wives and mistresses are common in monogamous societies, and 2) in reality, different moral standards apply to men and women in monogamous societies as well. Both of these are good points but this seems different from institutionalizing different moralities in relatively formal systems of justice and tradition...
For more than a century Christian missionaries preached, largely unsuccessfully, against polygyny. Through most of these years, polygyny rose in many places. However, now it seems to be declining, and my prediction is that while it is unlikely to ever disappear, its decline will continue. However, this has nothing to do with morality. Polygyny is ultimately rooted in the advantage large families have in many social/ecological settings. In an increasingly urbanized world, large families are a disadvantage to for most people. Without the motivation of wealth and power, polygyny is likely to diminish.