The "movement" against relativistic mass was started by Adler in 1987 with Does Mass Really Depend on Velocity, Dad? (his answer, "actually no, but don't tell your teacher"). It got a boost from Okun's two 1989 papers. In 1990 American Journal of Physics solicited Okun's contribution on the relativistic mass to be published alongside Sandin's, who defended it. Okun declined because he was helping Taylor and Wheeler write a dialog on the subject in the second addition of their book, and published a book of his own that addresses it.
Two decades later in The mass versus relativistic and rest masses Okun did not sound as optimistic as Physics SE:"The article “In defense of relativistic mass” by Sandin was published in 1991 and propagated the concept of relativistic mass for almost 20 years. In August 2008 I read a recent Wikipedia article “Mass in special relativity” which confronts the arguments of Adler and myself with arguments of Sandin. The unknown author in Wikipedia seems to share the views that relativistic mass and rest mass are ‘pedagogically useful’. The idea that according to the theory of relativity the mass of a body increases with its velocity is spread almost as widely as twenty years ago. It is promoted by the very popular books by Hawking and Jammer".
Here is the gist of Adler-Okun's argument as expressed by Okun in 2008:"The concept of relativistic mass, which increases with velocity, appears only in the framework of a language which is not compatible with the standard language of relativity theory and therefore impedes the understanding and learning the theory by beginners... The four equations... defining relativistic and rest masses completely ignore the space-time relativistic symmetry. Relativistic mass $m$ in these equations is a kind of a stub of a four-vector.These equations
are based on a naive but unfounded requirement that masses in relativity must have the property of additivity. They are based on the purely non-relativistic concept of mass as the measure of inertia. It is well known that in relativity this role is played not by mass but by energy".
He then traces the root of the problem to Einstein's famous equation:"Sandin keeps stressing that relativistic mass is the best concept for introducing relativity to ‘introductory students’. He thinks so because all of them know ‘the famous equation $E=mc^2$. This equation... is indeed deeply ingrained in the mass culture and widely believed to represent relativity theory. But unfortunately it misrepresents the standard theory of relativity as known to a majority of relativity experts. One consequence of the equation is the mass increasing with velocity. Though in his popular writings Einstein sometimes resorted to this equation, he carefully explained that ‘it is not good’ to introduce velocity dependent mass. He kept insisting that mass increases with energy but not with velocity and that this is expressed by the formula $E_0=mc^2$, where $E_0$ is the rest energy of a body".