Advances in high-performance computing make it possible to run atmospheric general circulation models (AGCMs) over an increasingly wider range of grid resolutions, using either globally uniform or variable-resolution grids. In principle, this is an exciting opportunity to resolve atmospheric process and scales in a global model and in unprecedented detail, but in practice, this grid flexibility is incompatible with the non- or weakly converging solutions with increasing horizontal resolution that have long characterized AGCMs. In the Community Atmosphere Model (CAM), there are robust sensitivities to horizontal resolution that have persisted since the model was first introduced over thirty years ago; the atmosphere progressively dries and becomes less cloudy with resolution, and parametrized deep convective precipitation decreases at the expense of stratiform precipitation. This study documents a convergence experiment using CAM, version 6 and argues that a unifying cause, the sensitivity of resolved dynamical modes to native grid resolution, feeds back into other model components and explains these robust sensitivities to resolution. The increasing magnitudes of resolved vertical velocities with resolution are shown to fit an analytic scaling derived for the equations of motion at hydrostatic scales. This trend in vertical velocities results in an increase in resolved upward moisture fluxes at cloud base, balanced by an increase in stratiform precipitation rates with resolution. Compensating, greater magnitude subsiding motion with resolution has previously been shown to dry out the atmosphere and reduce cloud cover. Here, it is shown that both the increase in condensational heating from stratiform cloud formation and greater subsidence drying contribute to an increase in atmospheric stability with resolution, reducing the activity of parametrized convection. The impact of changing the vertical velocity field with native grid resolution cannot be ignored in any effort to recover convergent solutions in AGCMs, and, in particular, the development of scale-aware physical parameterizations.