  Cobalt pyrochlore fluoride NaCaCo$_2$F$_7$ is a disordered frustrated magnet
composed of Co$^{2+}$ ions with an effective spin-$\frac{1}{2}$ magnetic moment
and exhibits spin freezing below $T_f \sim$2.4 K. We perform ultrasound
velocity measurements on a single crystal of the cubic NaCaCo$_2$F$_7$. The
temperature dependence of the bulk modulus (the breathing elastic mode)
exhibits Curie-type softening upon cooling below $\sim$20 K down to $T_f$,
which is suppressed by the magnetic field. This Curie-type softening should be
a precursor to the enhancement of the strength of exchange disorder via the
spin-lattice coupling, which causes the spin freezing. In contrast to the
magnetic-field-suppressed Curie-type softening in the bulk modulus, the
trigonal shear modulus exhibits softening with a characteristic minimum upon
cooling, which is enhanced by the magnetic field at temperatures below $\sim$20
K. This magnetic-field-enhanced elastic anomaly in the trigonal shear modulus
suggests a coupling of the lattice to the dynamical spin-cluster state. For
NaCaCo$_2$F$_7$, the observed elastic anomalies reveal an occurrence of
magnetic-field-induced crossover from an isostructural lattice instability
toward the spin freezing to a trigonal lattice instability arising from the
emergent dynamical spin-cluster state.
