  Semitoric systems are a type of four-dimensional integrable system for which
one of the integrals generates a global $S^1$-action; these systems were
classified by Pelayo and Vu Ngoc in terms of five symplectic invariants. We
introduce and study semitoric families, which are one-parameter families of
integrable systems with a fixed $S^1$-action that are semitoric for all but
finitely many values of the parameter, with the goal of developing a strategy
to find a semitoric system associated to a given partial list of semitoric
invariants. We also enumerate the possible behaviors of such families at the
parameter values for which they are not semitoric, providing examples
illustrating nearly all possible behaviors, which describes the possible limits
of semitoric systems with a fixed $S^1$-action. Furthermore, we introduce
natural notions of blowup and blowdown in this context, investigate how
semitoric families behave under these operations, and use this to prove that
each Hirzebruch surface admits a semitoric family with certain desirable
invariants; these families are related to the semitoric minimal model program.
Finally, we give several explicit semitoric families on the first and second
Hirzebruch surfaces showcasing various possible behaviors of such families
which include new semitoric systems.
