We report neutron scattering measurements of the structural correlations associated with the apparent relaxor transition in K$_{1-x}$Li$_x$TaO$_3$ for $x=0.02$ (KLT(0.02)). This compound displays a broad and frequency-dependent peak in the dielectric permittivity, which is the accepted hallmark of all relaxors. However, no evidence of elastic diffuse scattering or any soft mode anomaly is observed in KLT(0.02) [J. Wen et al., Phys. Rev. B 78, 144202 (2008)], a situation that diverges from that in other relaxors such as PbMg$_{1/3}$Nb$_{2/3}$O$_3$. We resolve this dichotomy by showing that the structural correlations associated with the transition in KLT(0.02) are purely dynamic at all temperatures, having a timescale on the order of $sim$THz. These fluctuations are overdamped, non-propagating, and spatially uncorrelated. Identical measurements made on pure KTaO$_3$ show that they are absent (within experimental error) in the undoped parent material. They exhibit a temperature dependence that correlates well with the dielectric response, which suggests that they are associated with local ferroelectric regions induced by the Li$^+$ doping. The ferroelectric transition that is induced by the introduction of Li$^+$ cations is therefore characterized by quasistatic fluctuations, which represents a stark contrast to the soft harmonic-mode-driven transition observed in conventional perovskite ferroelectrics like PbTiO$_3$. The dynamic, glass-like, structural correlations in KLT(0.02) are much faster than those measured in random-field-based lead-based relaxors, which exhibit a frequency scale of order of $sim$GHz and are comparatively better correlated spatially. Our results support the view that random fields give rise to the relaxor phenomena, and that the glass-like dynamics observed here characterize a nascent response.
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