Short answers, pulled from the story.
Constructivism was founded in 1915 by Vladimir Tatlin and Alexander Rodchenko. The term itself was first used in Naum Gabo's Realistic Manifesto of 1920, and Aleksei Gan fixed it as a book title in 1922.
Tatlin's Tower was Vladimir Tatlin's proposal for the Monument to the Third International, designed between 1919 and 1920 to include searchlights, projection screens, and moving components. It was never built because of a lack of money following the Russian Revolution.
Constructivists contributed directly to Soviet film: Rodchenko designed intertitles and animated sequences for Dziga Vertov's Kino Eye in 1924, and Aleksandra Ekster designed the sets and costumes for the science fiction film Aelita the same year. Vertov, Sergei Eisenstein, and documentarist Esfir Shub all understood their montage filmmaking as Constructivist practice.
The Left Front of the Arts organised Soviet Constructivists in the 1920s and produced the journal LEF, which ran from 1923 to 1925 and then continued as New LEF from 1927 to 1929. LEF defended the avant-garde against Socialist Realism and argued that cinema was more important than easel painting.
Naum Gabo established a version of Constructivism in England during the 1930s and 1940s, influencing artists including Victor Pasmore. Joaquín Torres García and Manuel Rendón spread the ideas across Europe and Latin America, reaching artists such as Oscar Niemeyer and Édgar Negret. A Constructivist International also met with Dadaists and De Stijl artists in Germany in 1922.
Leon Trotsky was the movement's main political patron, and after his expulsion along with the Left Opposition in 1927-28, Constructivism came under increasing suspicion. The Communist Party had questioned the use of government funds on avant-garde work as early as 1918, and by around 1934 the counter-doctrine of Socialist Realism was formally established in its place.