The concept of soliton management has been explored in the Bose-Einstein condensate and optical fibers. In this paper, our purpose is to investigate whether a similar concept exists for a variable-coefficient modified Korteweg-de Vries equation, which arises in the interfacial waves in two-layer liquid and Alfvén waves in a collisionless plasma. Through the Painlevé test, a generalized integrable form of such an equation has been constructed under the Painlevé constraints of the variable coefficients based on the symbolic computation. By virtue of the Ablowitz-Kaup-Newell-Segur system, a Lax pair with time-dependent nonisospectral flow of the integrable form has been established under the Lax constraints which appear to be more rigid than the Painlevé ones. Under such Lax constraints, multisoliton solutions for the completely integrable variable-coefficient modified Korteweg-de Vries equation have been derived via the Hirota bilinear method. Moreover, results show that the solitons and breathers with desired amplitude and width can be derived via the different choices of the variable coefficients.