The last gravitational waves emitted in the coalescence of two black holes are quasi-normal ringing modes of the merged remnant. In general relativity, the mass and the spin of the remnant black hole uniquely determine the frequency and damping time of each radiated mode. The amplitudes of these modes are determined by the mass ratio of the system and the geometry of the coalescence. This paper is part I of an analysis that aims to compute the "excitation factors" associated with misaligned binary black hole coalescence. To simplify the analysis, we consider a large mass ratio system consisting of a non-spinning body of mass $μ$ that inspirals on a quasi-circular trajectory into a Kerr black hole of mass $M$ and spin parameter $a$, with $μ/M \ll 1$. Our goal is to understand how different modes are excited as a function of the black hole spin $a$ and an angle $I$ which characterizes the misalignment of the orbit with the black hole's spin axis. Though the large mass ratio limit does not describe the binaries that are being observed by gravitational-wave detectors today, this limit makes it possible to quickly and easily explore the binary parameter space, and to develop insight into how the system's late ringing waves depend on the binary's geometry. In this first analysis, we develop the worldline which the small body follows as it inspirals and then plunges into the large black hole. Our analysis generalizes earlier work by Ori and Thorne to describe how a non-equatorial circular inspiral transitions into a plunging trajectory that falls into the black hole. The worldlines which we develop here are used in part II as input to a time-domain black hole perturbation solver. This solver computes the gravitational waves generated by such inspirals and plunges, making it possible to characterize the modes which the coalescence excites.

18 pages, 7 figures. This update matches the text in the final published version