<p>In 2021, the Luchko operational calculus has been proposed for general fractional derivatives (GFDs) with Sonin kernels. This operational calculus is a generalization of the Mikusinski operational calculus for the integer-order derivatives and the operational calculi of fractional derivatives constructed by Luchko and co-authors since 1993. In this paper, we propose the extension of the Luchko operational calculus from single-kernel GFDs to multi-kernel GFDs. We prove the first and second fundamental theorems of general fractional calculus for multi-kernel GFDs of the Caputo type. For the multi-kernel GFDs of the Caputo type, we formulate a multi-kernel extension of the Luchko operational calculus to solve equations describing a wider class of non-localities in space and time. An extension of the Luchko operational calculus to multi-kernel general fractional integro-differential operators, which are not GFDs in general, is also proposed. Using this multi-kernel extension of operational calculus, we derive solutions of fractional differential equations with multi-kernel GFDs and multi-kernel general fractional integro-differential operators. Examples of such solutions are proposed for equations with multi-kernel operators whose kernels are different Sonin kernels or the Laplace product of different Sonin kernels.</p>