We study the parametrized Hamiltonian action functional for
finite-dimensional families of Hamiltonians. We show that the linearized
operator for the $L^2$-gradient lines is Fredholm and surjective, for a generic
choice of Hamiltonian and almost complex structure. We also establish the
Fredholm property and transversality for generic $S^1$-invariant families of
Hamiltonians and almost complex structures, parametrized by odd-dimensional
spheres. This is a foundational result used to define $S^1$-equivariant Floer
homology.