To determine which statement about the tyrants of Greek city-states in the mid-600s BCE is not correct, let's analyze each option:
A. Tyrants frequently established dynasties that ruled until they were overthrown by the people - This is generally true; many tyrants did establish dynasties, and some were indeed overthrown by the populace.
B. Tyrants were often supported by the middle class who wanted political reform - This is also accurate; many tyrants gained support from the middle class and other groups seeking change from the aristocratic rule.
C. Wealthy and aristocratic citizens usually lost influence when a new tyrant took power - This is true as well; tyrants often diminished the power of the aristocracy to solidify their own rule.
D. Many tyrants were military leaders who promised to extend more rights to citizens - While some tyrants may have been military leaders, the statement that they promised to extend rights is somewhat misleading as many did not prioritize extending rights to citizens and instead focused on consolidating their own power.
Given the analysis, D is the statement that can be considered not entirely correct, as the extent to which tyrants promised to extend rights varied widely and was not a defining characteristic of their rule.