\section{Introduction and contribution}\label{sec:introduction}
Regularized immersed boundary (IB) methods couple a fluid velocity on a background grid to an independently represented interface \citep{peskin2002}. Part~I of this series \citep{parti2026} analyzed a conforming realization under dynamic Cartesian refinement for fixed geometric interfaces, including a circular disk with prescribed angular velocity. This paper allows the rigid interface itself to translate and rotate. In particular, rotation of a noncircular body changes the constrained velocity space even if the Cartesian mesh is left unchanged.

The required extension is not obtained by flattening a neighborhood of the entire swept interface. Such a neighborhood need not shrink to the instantaneous boundary and may exclude legitimate fluid velocities. Instead, our comparison function is flattened near the current interface and changes continuously in time. Its Eulerian time derivative generally fails to approximate the exact derivative in $H^1$. We prove that its error does tend to zero in the bulk $L^2$ norm, which is sufficient for the energy argument. Orthogonality against the current test space then removes an explicit sum of mesh-change defects, even though the previous comparison function is not in the current constraint space.

There is substantial prior convergence theory for IB methods. Mori established velocity convergence for periodic Stokes flow and a small-amplitude dynamic problem \citep{mori2008}; Liu and Mori analyzed discrete-delta accuracy and stationary Stokes velocity and pressure convergence \citep{liumori2012,liumori2014}. Li, Pan, and Ruiz-\'{A}lvarez treated stationary Stokes discretizations with prescribed outer boundary velocity \citep{lipan2025}. The moving-particle analysis of Milewski and Peskin \citep{milewski2025} concerns a different, codimension-two elastic coupling with a fixed physical mollifier width. These results do not constitute a convergence proof for the present moving codimension-one velocity constraints with shrinking regularization and changing Cartesian enrichments.

Prescribed rigid motion in fictitious-domain multiplier formulations is established \citep{glowinski1997}, as are constrained rigid-body IB operators \citep{kallemov2016}. Adaptive IB implementations include the work of Roma, Peskin, and Berger \citep{roma1999} and the dynamically adaptive DLM--IB framework of Zeng, Bhalla, and Shen \citep{zeng2022}. Regularization estimates and adaptive regularized elliptic methods provide neighboring analytical tools \citep{heltai2020,heltai2023}; optimal adaptive fictitious-domain convergence is also known in an elliptic setting \citep{berrone2019}. The present theorem is for a specified conforming method, not a proof for these fluid solvers or an adaptive-optimality result.

Moving-domain Stokes stability and convergence also predate this work. Eulerian unfitted methods extend previous-time values to newly occupied fluid regions \citep{lehrenfeld2019}. Burman, Frei, and Massing analyze stabilized Eulerian time stepping for moving-domain Stokes flow \citep{burman2022moving}; von Wahl, Richter, and Lehrenfeld study a related unfitted mixed method including geometric approximation \citep{vonwahl2022}. Evolving-space theory identifies the transport and compatibility questions of time-dependent function spaces \citep{alphonse2015,djurdjevac2023}. ALE Stokes error analysis includes optimal convergence estimates \citep{rao2025}, and recent Piola-based space-time DG work treats divergence-free moving-domain spaces without spatial/time partition restrictions \citep{beirao2026}. Our method stays in physical Eulerian coordinates and imposes regularized surface moments on global background velocities. The claimed contribution is the concrete moving-collar recovery and its use with arbitrarily changing retained-coarse Cartesian spaces, not a first convergence theorem for moving domains. Likewise, stable mesh transfer and projection-change errors are established topics \citep{brenner2014,lundquist2021,gawlik2015}.

Computational IB studies motivate the interface and motion distinctions. He et al.\ \citep{he2022multiphase} introduced pressure-boundary treatment to reduce spurious penetration through thin structures in two-phase flow. The prescribed heave and pitch studied by Zheng et al.\ \citep{zheng2024swimmers} genuinely change the immersed plate's geometry, unlike fixed-disk rotation. Their free surface and nonlinear flow are beyond the present Stokes model. Fixed-geometry IB applications include backward-facing-step LES \citep{yang2021step} and airfoil flow-transition calculations \citep{chen2025airfoil}; flexible-canopy simulations \citep{he2022canopy} illustrate the additional structural dynamics excluded from this rigid-body series. These method and application connections do not serve as mathematical premises for the proofs.

The analysis makes four structural choices explicit. Velocities are curls of conforming Cartesian bicubic Hermite streamfunctions. The full coarse space is retained while local enrichments may be added or removed at every step. Interface constraints are exact surface moments of an exact convolution. The continuous target is a sufficiently regular Eulerian Stokes solution with a square-integrable interface reaction and a prescribed interior extension. Under these assumptions we prove feasibility, unconditional driven stability, moving-interface slip control, and an energy-error estimate independent of the number of refinement events. The square-root temporal term accounts for interface displacement and piecewise smooth fields; it is not an assertion that backward Euler cannot converge faster on smoother problems.