Let \(D([0,T]; R^d)\) be a Skorokhod space, \(\mu^\varepsilon(A)=P\{\xi^\varepsilon(\cdot)\in A\}\), \(A\in {\mathcal B}(D([0,T]; R^d))\), \(\varepsilon > 0\), be a family of probability measures, corresponding to the \(d\)-dimensional locally infinitely divisible processes \(\xi^\varepsilon(t)\), \(t\geq 0\), \(\varepsilon > 0\), defined on some filtered probability space. A general principle of large deviations is proved for the family \(\{\mu^\varepsilon\), \(\varepsilon>0\}\) in terms of the local characteristics of \(\xi^\varepsilon\), \(\varepsilon>0\). Some special cases are discussed in detail.