Indeed they did: civil state-to-state relations were generally maintained through the cold war, especially in such moments, but this time it seems to have been genuinely heartfelt following the relaxation of tension between the superpowers since the missile crisis thirteen months earlier:

MOSCOW, Nov. 23, 1963 (UPI) - Soviet Premier Nikita S. Khrushchev said today the death of President Kennedy "is a heavy blow to all people who hold dear the cause of peace and Soviet-American cooperation." In a cable to President Johnson, Khrushchev expressed the "indignation of Soviet people against the culprits of this base crime."

Khrushchev described the late president as a "person of broad outlook who realistically assessed the situation and tried to find ways to negotiate settlements of the international problems which now divide the world."

"The Soviet government and the Soviet people share the deep grief of the American people over this great loss and express the hope that the search for setline the disputable issues, a search to which President J.F. Kennedy made a tangible contribution, would be continued in the interests of peace, for the benefit of mankind," he said.

A 1966 FBI memo reported on Soviet concerns in the wake of the killing:

A source who has furnished reliable information in the past and who was in Russia on the date of the assassination of the late President John F. Kennedy advised on December 4, 1963, that the news of the assassination of President Kennedy was flashed to the Soviet people almost immediately after its occurrence. It was greeted by great shock and consternation and church bells were tolled in the memory of President Kennedy.

According to our source, officials of the Communist Party of the Soviet Union believed there was some well-organized conspiracy on the part of the "ultraright" in the United States to effect a "coup." They seemed convinced that the assassination was not the deed of one man, but that it rose out of a carefully planned campaign in which several people played a part. They felt those elements interested in utilizing the assassination and playing on anticommunist sentiments in the United States would then utilize this act to stop negotiations with the Soviet Union, attack Cuba and thereafter spread the war. As a result of these feelings, the Soviet Union immediately went into a state of national alert.

Khrushchev in a 1964 message to Robert Kennedy for the JFK library noted the late President's June 1963 American University address as suggestive of greater openness to pragmatic co-existence:

That statement can be called courageous and more realistic than what the Soviet Union and other countries of the socialist world often heard from American shores. Although that statement of the President of the United States, too, had some conflicting points and a tribute unfortunately was paid in it to the so‐called policy of ‘containment and pushing back of Communism’, as a whole, however, it proceeded from acknowledgement of the inevitability and necessity of coexistence of states with different social systems....

Although everyone knows that President John F. Kennedy was a man of an ideology which is opposite to ours, and represented interests of a state standing on a political platform which is opposite to ours, nevertheless in that statement of his he was already outlining more realistic principles of the relations between countries with different social systems and thus, if his idea is broadened, between two social systems.