ALMA-IMF XIII: N$_2$H$^+$ kinematic analysis on the intermediate protocluster G353.41

The ALMA-IMF Large Program provides multi-tracer observations of 15 Galactic massive protoclusters at matched sensitivity and spatial resolution. We focus on the dense gas kinematics of the G353.41 protocluster traced by N$_2$H$^+$ (1$-$0), with an spatial resolution $\sim$0.02 pc. G353.41, at a distance of $\sim$2 kpc, has a mass of $\sim$2500 M$_{\odot}$ within $1.3\times1.3$ pc$^2$. We extract the N$_2$H$^+$ isolated line component and we decompose it by fitting up to 3 Gaussian velocity components. This allows us to identify velocity structures that are impossible to identify in the traditional position-velocity diagram. We identify multiple velocity gradients on large ($\sim$1 pc) and small scales ($\sim$0.2 pc). We find good agreement between the N$_2$H$^+$ velocities and the previously reported DCN core velocities, suggesting that cores are kinematically coupled to the dense gas in which they form. We measure 9 converging ``V-shaped'' velocity gradients ($\sim20$ km/s/pc), located in filaments, which are sometimes associated with cores near their point of convergence. The average inflow timescale is $\sim67$ kyr, or about twice the free-fall time of cores in the same area ($\sim33$ kyr) but substantially shorter than protostar lifetime estimates ($\sim$0.5 Myr). We derive mass accretion rates in the range of $(0.35-8.77)\,\times\,10^{-4}$ M$_{\odot}$/yr. This feeding might lead to further filament collapse and formation of new cores. We suggest that the protocluster is collapsing on large scales, but the velocity signature of collapse is slow compared to pure free-fall. These data are consistent with a comparatively slow global protocluster contraction under gravity, and faster core formation within, suggesting the formation of multiple generations of stars over the protocluster lifetime.