We consider the Markov process defined by some pseudo-differential operator of the order $1<α<2$ as the process generator. Using a pseudo-gradient operator, that is, the operator defined by the symbol $iλ|λ|^{β-1}$ with some $0<β<α$, the perturbation of the Markov process by the pseudo-gradient with a multiplier integrable at some great enough power is constructed. Such perturbation defines a family of evolution operators, the properties of which are investigated.