Reconstruction of the oldest surviving text in a Finno-Ugric language: The Funeral Sermon (Halotti beszéd) written between 1192–1195 from the Codex Pray . Reconstruction by Dezső Pais renowned Hungarian linguist and member of the Hungarian Academy of Sciences (MTA). Read by GyörgyBánffy.
Well I am not a linguist, but I'll attempt to answer your question according to the best of my knowledge. I would say the greatest change was the language reformation of the late 18th and early 19th centuries. This visibly increased the vocabulary and created rules and regulations to confine the language. Another influence that should also be mentioned is borrowings from Latin and much later German.
About the second part of your question, quiet interestingly the earliest Hungarian sentence from an earlier document from the 11th century is showing an example of morpheme separation: "rea meneh" which today is written as "rámenő". Otherwise this is an exception and Hungarian is a very synthetic language, with some very extreme cases. The "Funeral sermon" which I posted here does not show cases of isolated morphemes. Hungarian being an agglutinative language uses inflections, but statistically speaking I wouldn't be able to tell you weather the rate of inflection has changed in percentage or not in the past 800 or so years.
What were the main trends of linguistic change in Hungarian language from 12 to 21 centuries? Have it become more analytical and inflectional?