We present a preliminary analysis of the effect of dynamical friction on the orbits of part of the globular clusters in our Galaxy. Our study considers an anisotropic velocity dispersion field approximated using the results of studies in the literature. An axisymmetric Galactic model with mass components consisting of a disc, a bulge, and a dark halo is employed in the computations. We provide a method to compute the dynamical friction acceleration in ellipsoidal, oblate, and prolate velocity distribution functions with similar density in velocity space. Orbital properties, such as mean time-variations of perigalactic and apogalactic distances, energy, and z-component of angular momentum, are obtained for globular clusters lying in the Galactic region $R \lesssim$ 10 kpc, $|z| \lesssim$ 5 kpc, with $R,z$ cylindrical coordinates. These include clusters in prograde and retrograde orbital motion. Several clusters are strongly affected by dynamical friction, in particular Liller 1, Terzan 4, Terzan 5, NGC 6440, and NGC 6553, which lie in the Galactic inner region. We comment on the more relevant implications of our results on the dynamics of Galactic globular clusters, such as their possible misclassification between the categories 'halo', 'bulge', and 'thick disc', the resulting biasing of globular-cluster samples, the possible incorrect association of the globulars with their parent dwarf galaxies for accretion events, and the possible formation of 'nuclear star clusters'.