The Rabam, Mishneh Torah- Zmanim- Hilchot Shofar, Sukkah and Lulav 7:13-18 speaks about the evolvement of the mitzvah of Lulav:
The Original Mitzvah:
The mitzvah of shaking the lulav only applies on the first day of the holiday of Sukkot in every place during every age and is practiced even on Shabbat, as we see in Vayikra 23:40 “And on the first day, you shall take for yourselves fruit of the majestic tree, branches of palm trees, boughs of the leafy tree, and willows of the brook…”
In the Mikdash (Jerusalem) alone the lulav is taken on each of the seven days of the holiday as it says “and rejoice before the Lord your God for seven days.” When Shabbat falls during the later days, the lulav is not taken. This is a decree lest one carry it four cubits in the public domain, as decreed regarding the shofar.
Why was this decree not put into effect on the first day of Sukkot? Because taking the lulav on the first day is a mitzvah from the Torah, even outside of Jerusalem. Thus, the laws applying to it are not the same as those applying to the remaining days since on the subsequent days of the holiday a person is obligated to take the lulav only in the Mikdash (Jerusalem).
Changes once the Beit HaMikdash (Temple) was Destroyed:
When the Beit HaMikdash was destroyed, the Sages ordained that the lulav be taken everywhere for the entire seven days of the festival, as a remembrance. Every day the blessing would be recited over it. This enactment, like the other enactments instituted by Rabbi Yochanan ben Zakkai when the Temple was destroyed, is only temporary. When the Temple is rebuilt, these matters will return to their original status.
While the Temple was standing, the lulav would be taken in Jerusalem when the first day of Sukkot fell on Shabbat. The same applies in other places (throughout Eretz Yisrael and the Diaspora) where they were certain that this day (the fifteenth of Tishrei) was celebrated as a holiday in Eretz Yisrael. However, the places which were distantly removed from Jerusalem would not take the lulav on this day because of the doubt.
When the Beit HaMikdash was destroyed, the Sages forbade everyone from shaking the lulav when the first day of the holiday occurred on Shabbat including the inhabitants of Eretz Yisrael who had sanctified the new moon. This was instituted because of the inhabitants of the distant settlements, who were not aware of when the new month had been declared. Thus, a uniform guideline was established, rather than having some take the lulav on the Shabbat and some not. The guiding principle was that the obligation of taking the lulav on the first day applies in all places and there is no longer a Temple to use as a point of distinction.
At present, when everyone follows a fixed calendar, the matter remains as it was, and the lulav is not taken on Shabbat anywhere- not in the outlying territories or in Eretz Yisrael even on the first day of the festival. This applies even though everyone knows the actual day of the month. As stated above, the reason for the prohibition of taking the lulav on Shabbat is a decree lest one carry it four cubits in the public domain.
We learn from the Rambam that in our time, nobody shakes the lulav on Shabbat regardless if they are in Jerusalem, Israel or the Diaspora. When the Beit HaMikdash will be rebuilt, we will go back to shaking the lulav on the first day, even on Shabbat but the Diaspora will not shake to lulav on the other days of the holiday. Hopefully, when the mashiach comes all of the Jews will be back in Israel and then they can have the opportunity to come to Jeusalem and shake the lulav there.
Until that time, Jews all over the world will shake the lulav on every day of the holiday aside from Shabbat and yearn for the Beit HaMikdash to be rebuilt speedily in our days.