We have some such cases, but not only are they more limited in number but also, at times, they differ from "communist-to-capitalist" cases qualitatively as well. In cases where people left the Western bloc for the USSR, ideology seems to have played a key role more often than not, whereas escapes from communist countries to the capitalist bloc were often less ideologically and more economically related, or perhaps more related to the hardships of daily life perceived by people who had lived in communist countries before their emigration.

An interesting case is that of the Cambridge Five. The Cambridge Five was the name of a group of British intelligence operatives who also worked as double agents for the KGB. Two members of the spy ring defected to the USSR in 1951, another member of the Cambridge Five named Kim Philby followed them in 1963. A similar case was that of William Martin and Bernon Mitchell, two American cryptologists who had worked for the NSA until 1960 when they defected to the USSR for similar reasons to the Cambridge Five.

But there were "ordinary people" who fled the capitalist countries for the USSR too. There were people who left the US during the Great Depression, presumably impressed with the relative stability of the USSR as perceived by the contemporaries back then. There where also people influenced by socialist or communist propaganda emigrated to the USSR and Tim Tzouliadis wrote a book exploring mainly such cases called The Forsaken.

Emigration, of course, was not limited to the UK and the US. You might also want to read this article by the Finnish historian Auvo Kostiainen where he investigates illegal emigration from Finland to the USSR in the 1930s: https://www.genealogia.fi/emi/art/article237e.htm

To sum up, we have many cases of emigration from the capitalist bloc to the USSR indeed. But such cases are vastly outnumbered by the other side of this emigration traffic. Moreover, they often came in a more ideological package, be it double agent defections or ideologically-motivated emigrations.

There a many cases of African American radicals fleeing/departing the United States for communist countries like Cuba, China, and North Korea during the Cold War. The most famous case is probably Robert and Maisie Williams who were prominent Civil Rights activists during the 1960s. Originally a leading figure in the NAACP's Monroe, NC branch, the Williams' grew dissatisfied with the organization's nonviolent approach in resisting organizations like the KKK and became prominent advocates of armed resistance to white supremacy, establishing the Black Armed Guard. Williams was eventually charged with kidnapping in 1961 and fled first to Cuba where he published a radical black newspaper and later China.

This is a growing subfield within the Cold War historiography. The best book on the subject (and an amazing book overall) is Taj Robeson Frazier's The East is Black: Cold War China in the Black Radical Imagination (2014), though Judy Tzu-Chun Wu's Radicals on the Road: Internationalism, Orientalism, and Feminism during the Vietnam Era (2013) is also quite good. Keisha Blain's work on black internationalist women may also be of interest, though it is focused less on the Cold War period.