Is Halacha (specifically Halacha, not minhag) capable of changing, or is it an immutable constant? In other words, is Halacha an inflexible set of rules that never bend to societal needs, scientific findings, new readings/interpretations of written or oral law etc? Are there specific examples of Halacha directly changing?
Short answer: The Written Torah does not and has not changed. The Oral Torah only changes under very limited circumstances.
Background: According to all opinions, the Written Torah (i.e. the Sefer Torah on klaf, the Chumash, 5 Books of Moses) was received directly from Hashem, and is the same as the one we have today. All Sifrei Torah in existence from all parts of the Jewish people, including areas which were not in contact with each other for millenia, match with the exception of some questionable חסר\מלא readings (vowels) and one question of an א vs. a ה (the verse פצוע דקה\דקא). Both Maimonides in his 13 Principles and Rav Yosef Albo (who attacks Maimonides for having so many Principles) in Sefer HaIkkarim consider acceptance of the legitimacy/inerrancy of the Written Torah as a core tenet of Judaism.
The Written Torah empowers the Sages to decide questionable cases in halacha in Deuteronomy 17:8: "When a matter of judgement is too difficult for you, between blood and blood, between law and law, you shall stand up and go up to the place which Hashem your God has chosen." The Sanhedrin of 70 Sages was established by Moshe, and decides cases by majority rule (Exodus 23:2). A centralized Sanhedrin remained in operation through the end of the period of the Gemara, circa 500CE, and the decisions of that Sanhedrin are binding on all Jews (Maimonides, Preface to the Code). The Sanhedrin was also empowered to enact Rabbinical decisions to strengthen and protect the borders of Torah Law (Deuteronomy 17:1). During history, the Sanhedrin occasionally made new enactments (mentioned several places in the Gemara) and rarely even changed its mind, which it can only do if it is greater in wisdom and numbers than the prior court (Eduyot 1:5).
After the end of the period of the Gemara, due to political changes, for approximately 1000 years Jewish communities were largely out of touch with each other. Initially communities outside of Babylon remained in touch with the central yeshivot of the Geonim and received halachik instruction from them, but in the Middle Ages communication broke down further and individual Rabbonim in local communities decided questions and wrote works of commentary, novellae, and responsa. The Gemara itself does not always resolve questions conclusively, and many different halachik traditions developed - some of which may reflect differences between Babylonian and Israeli Jewry, or other differences which predate the sealing of the Gemara.
In the early 16th century, Rabbi Yosef Karo (in Turkey and then Tzfat) attempted to consolidate all opinions of major authorities (Rishonim, "the prior ones") in his monumental work Beit Yosef, then decide all questions by going after the majority. At the same time in Krakow, Rabbi Moshe Isserles ("the Rema") was writing a similar work which codified Ashkenazi opinion, but instead based on the opinion of the most recent authorities, on the grounds that they had seen and considered any prior opinions. Rabbi Karo ("The Mechaber"), published his work first in concise form, the Shulchan Aruch ("set table"), and it was accepted as definitive by North African and Middle Eastern Jews. The Rema published his work as a commentary to the Shulchan Aruch, and it too was accepted by Ashkenazi Jews. As a result, both groups ended up studying the same unified work, which gained thereby universal standing (with some exceptions).
There are some exceptions to this: during the same period, another major Ashkenazi authority, Rabbi Shlomo Luria ("Maharshal") rejected both approaches in favor of deciding halacha based on pure logic. Many of the Maharshal's rulings (which seem to always be stringencies) were incorporated into later commentaries on the Shulchan Aruch, which proliferated, sometimes arguing with the Machaber and Rema, sometimes considering new cases. A large group of Yemenite Jews accepted Maimonides as their authority during his lifetime, and continue to act only in accordance with his rulings to this day. More recently, students of the Yeshivat Brisk reject the basis of the Mechaber and Rema's concept of deciding between Rishonim, and attempt to fulfill all opinions.
Broadly speaking, conducting oneself in accordance with a given opinion is a vow. An individual who accepts a stringency may be able to nullify that vow; a community cannot do so (see R'Moshe Feinstein, OC 2.21 & 4.34, YD 1.13 & 2.15). In addition, authorities will not argue on other authorities who predate their own period (i.e. authorities today can argue, but they do not argue with Rishonim). As a result, precedent in Halacha is largely binding, even if in theory there would be both reason and space to look at a given issue in a different light and reach different conclusions. Cases of mistake or mistaken acceptance of practice ("I thought everyone kept this stringency!") are more readily nullified.