Kinematic properties, such as the vorticity, divergence, and rate of strain, describe the evolution of velocity and encapsulate the rate of deformation and rotation of a fluid parcel. Kinematic properties are particularly important in submesoscale flows, where the Rossby number $Ro$ becomes $\mathcal{O}$(1). However, since submesoscale flows are typically highly anisotropic, evolving over timescales of hours to days and over length scales of $\mathcal{O}$(0.1-20) km, their velocity and velocity gradients are challenging to observe from contemporary measurement platforms. With increasing quantity and quality of Lagrangian drifter observations, we here study the velocity gradient estimation from swarms of drifters. First, by simulating drifter swarms, we quantify the sources of uncertainty in the velocity gradient calculation associated with the deformation of drifter swarms using a bootstrap approach and determine the ideal parameter space for the application to observed trajectories. We then apply the most robust method - a two-dimensional, linear least-squares fit of the swarm velocity field - to a drifter dataset from the Bay of Bengal. The drifter-estimated vorticity, divergence, and lateral strain rate reflect the presence of a cyclonic mesoscale eddy, frontal circulation as well as banded patterns that are likely generated by turbulent thermal wind. The distributions and magnitudes of the kinematic properties suggest the presence of submesoscale flows associated with a strong freshwater-dominated density front. Understanding and improving methods for multi-drifter observations are timely challenges which will help design future drifter experiments with the goal of observing two-dimensional divergence and vorticity in submesoscale flows.