Something which will destabilize very easily.
The Birlings (an inspector calls) were a ____ family.
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dependent on circumstances beyond one's control; uncertain; unstable; insecure: a precarious livelihood.
dependent on the will or pleasure of another; liable to be withdrawn or lost at the will of another:
changeable; mercurial; flighty: a volatile disposition.
Does the word "unstable" not suit your needs?
prone to change, fail, or give way; not stable. —Oxford Dictionaries
apt or likely to change.
The family might be prone to labile moods From Psychcentral:
Marked fluctuation of mood; can be a symptom of a disorder such as schizophrenia.
Example: A woman seeks help for “ups and downs” lasting days or weeks; she can’t seem to keep her emotions on an even keel.