Low blood sugar can happen after a spike, but not always. It depends on how much insulin production the beta cells in your pancreas are still capable of producing. If the body can't produce enough insulin to lower the blood sugar after a meal or snack is eaten, blood sugar levels continue to rise. Low blood sugars can also occur when insulin production is more than what was needed for what you ate or not enough food was eaten in proportion to the amount of medication in the bloodstream.
In a non-diabetic person if blood sugar starts to drop and no carbohydrates have been eaten to bring it back up again, gluconeogenesis kicks in converting protein to glucose in the liver if there are no glycogen stores in the liver (the body goes for the glycogen stores first). This is a slow process, however and not like eating a meal or snack.
Unstable blood sugars can happen for different reasons, some of them the same mechanisms as above (too much insulin/not enough food, too much food/not enough insulin) or just because the body is going through a repeated cycle of high carb/elevated blood sugar/increased insulin production/blood sugar crash.

Hope this hasn't confused you totally!