Question: During periods of time, this acquistion or imposition of elements of Polish culture, such as the language, under populations of territories controlled, or under the influence of Poland; Jan Kazimierz Chodkiewicz, a Lithuanian nobleman, is a prime example of this act - what is it called? 

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Malayisation: Malayisation is a process of assimilation and acculturation, that involves acquisition (Malay: "Masuk Melayu" , literally "embracing Malayness") or imposition (Malay: "Pemelayuan" or "Melayuisasi") of elements of Malay culture, in particular, Islam and Malay language, as experienced by non-Malay populations of territories controlled or substantially influenced by historical Malay sultanates and modern .  It is often described as a process of civilisational expansion, drawing a wide range of indigenous peoples into the Muslim, Malay-speaking polities of Maritime Southeast Asia.

Jan Kazimierz Chodkiewicz: Jan Kazimierz Chodkiewicz (1616–1660) (Lithuanian: "Chodkevičius" ) was a Polonized Lithuanian nobleman (szlachcic).

Chodkiewicz: The Chodkiewicz (Belarusian: Хадкевіч , Polish: "Chodkiewiczowie" , Lithuanian: "Chodkevičius, Kadkevičius, Katkevičius, Katkus" ) family was one of the most influential noble families of Ruthenian descent in the Polish-Lithuanian Commonwealth in the 16th–17th centuries.  Chodko Jurewicz, chamberlain to Grand Duke Vytenis, was probably the ancestor of the whole clan and gave it the name "Chodkiewicz", meaning "son of Chodzko".  Surnames were not used in that time, but apparently later in history, the name "Chodzko" became a surname after Christianization of Chodzko Juriewicz, father of Iwan (Jan) Chodkiewicz.  They bore the Chodkiewicz coat of arms.  In 1572, Jan Hieronimowicz Chodkiewicz abandoned Calvinism for the Roman catholicism with his two sons, which made them first polonized generation of the once Ruthenian family.  Emperor Charles V granted the title of Count of the Roman Empire to the whole family.

Aleksander Chodkiewicz: Aleksander Chodkiewicz (Lithuanian: "Aleksandras Chodkevičius" , Belarusian: Аляксандар Хадкевіч ; ca. 1475 – 28 May 1549) was a noble from the Grand Duchy of Lithuania, founder of the Supraśl Orthodox Monastery.  He inherited vast possessions from his father Ivan Chodkiewicz, which made him 11th wealthiest person in the Grand Duchy according to the military census of 1528.  Via his mother Jawnuta (Agnieszka) of the Belsky family, he was second cousin to Kings of Poland John I Albert, Alexander Jagiellon, and Sigismund I the Old.  Chodkiewicz quickly gained influence under Alexander Jagiellon, becoming royal marshal.  However, after Alexander's's death in 1506 his career stagnated.  He supported Queen Bona Sforza, gaining her favor and becoming starost of Brest in 1528.  The peak of his career came in 1544 when young Sigismund II Augustus, still influenced by his mother, appointed Chodkiewicz as voivode of Nowogródek and his son Hieronim as castellan of Trakai.

John II Casimir Vasa: John II Casimir (Polish: "Jan II Kazimierz Waza" ; German: "Johann II.  Kasimir Wasa" ; Lithuanian: "Jonas Kazimieras Vaza" ; 22 March 1609 – 16 December 1672) was King of Poland and Grand Duke of Lithuania during the era of the Polish–Lithuanian Commonwealth, Duke of Opole in Upper Silesia, and titular King of Sweden 1648–1660.  In Poland, he is known and commonly referred as Jan Kazimierz.  His parents were Sigismund III Vasa (1566–1632) and Constance of Austria (1588–1631).  His older brother, and predecessor on the throne, was Władysław IV Vasa.

History of Poland during the Jagiellonian dynasty: The rule of the Jagiellonian dynasty in Poland between 1386 and 1572 spans the late Middle Ages and early Modern Era in European history.  The dynasty was founded by the Lithuanian Grand Duke Jogaila (Władysław II Jagiełło), whose marriage to Queen Jadwiga of Poland formed a Polish–Lithuanian union.  The partnership brought vast territories controlled by the Grand Duchy of Lithuania into Poland's sphere of influence and proved beneficial for both the Polish and Lithuanian people, who coexisted and cooperated in one of the largest political entities in Europe for the next four centuries.

Polonization: Polonization (or Polonisation; Polish: "polonizacja" ) was the acquisition or imposition of elements of Polish culture, in particular the Polish language, as experienced in some historic periods by the non-Polish populations of territories controlled or substantially under the influence of Poland.  As with other examples of cultural assimilation, it could either be voluntary or forced and is most visible in the case of territories where the Polish language or culture were dominant or where their adoption could result in increased prestige or social status, as was the case of the nobility of Ruthenia and Lithuania.  To a certain extent Polonization was also administratively promoted by the authorities, particularly in the period following World War II.

Roman Dmowski: Roman Stanisław Dmowski (9 August 1864 – 2 January 1939) was a Polish politician, statesman, and co-founder and chief ideologue of the right-wing National Democracy ("ND": in Polish, ""Endecja"") political movement.  He saw the aggressive Germanization of Polish territories controlled by the German Empire as the major threat to Polish culture and therefore advocated a degree of accommodation with another power that had partitioned Poland, the Russian Empire.  He favored the re-establishment of Polish independence by nonviolent means, and supported policies favorable to the Polish middle class.  During World War I, in Paris, through his Polish National Committee he was a prominent spokesman, to the Allies, for Polish aspirations.  He was a principal figure instrumental in the postwar restoration of Poland's independent existence.

Jan Karol Chodkiewicz: Jan Karol Chodkiewicz ("c".  1560 – 24 September 1621; Belarusian: Ян Караль Хадкевіч, "Jan Karal Chadkievič" , Lithuanian: "Jonas Karolis Chodkevičius" ) was a military commander of the Polish–Lithuanian Commonwealth army who was from 1601 Field Hetman of Lithuania, and from 1605 Grand Hetman of Lithuania, and was one of the most prominent noblemen and military commanders of the Polish–Lithuanian Commonwealth of his era.  His coat of arms was Chodkiewicz, as was his family name.

Jan Klemens Branicki: Count Jan Klemens Branicki (also known as Jan Kazimierz Branicki) (21 September 1689 – 9 October 1771) was a Polish nobleman, magnate and Hetman, Field Crown Hetman of the Polish-Lithuanian Commonwealth between 1735 and 1752, and Great Crown Hetman between 1752 and 1771.  One of the wealthiest Polish magnates in the 17th century, owner of 12 towns, 257 villages and 17 palaces.  He was the last male representative of the Branicki family.

Supporting paragraphs: Jan Kazimierz Chodkiewicz Polonization
Answer: Polonization