Hitler envisioned a thousand year Reich. How did he plan for power to shift during these years after his death? Would it be dynastical, inherited by future children like a Feudal Kingdom/Empire, or "democratically" between senior officials of the Party voting for a new Führer?

Hitler's official successor through out the war until 1945 was Herman Goering, who was authorized to take over if Hitler was killed or "incapacitated" during the war. This changed near the middle of 1945, when the Red Army closed in on Berlin. Goering assumed that Hitler's decision was to stay in the bunker made him unable to rule (incapacitated). Goering sent a telegram asking to take over governance of what was left of the Reich. Martin Bormann, Hitler's secretary and leader of the actual Nazi party made it seem like Goering was launching a coup (Bormann hated Goering), Hitler expelled Goering from the party (along with Himmler who got caught negotiating with the Western allies) and revoked the decree naming Goering as Hitler's successor.

This technically doesn't answer the question, which I think is dealing more with long term succession but what Hitler did after revoking the decree naming Goering as his successor is worth noting and I think gives a better glimpse into a post Hitler Nazi-Germany; mainly because AFAIK Hitler didn't really leave give any sort of idea as to what the post-Hitler Nazi government would look like exactly, preferring instead to focus on the nation itself (ie. territory, alliances, etc.)

After Goering's fall from grace, Hitler drew up a new will and nominated new successors. He abolished the post of Fuhrer and nominated Goebbels and Donitz to serve as Reich Chancellor and Reich President respectively.

So where as Goering's succession was meant as an emergency act intended to get someone reliable in power to continue the war if Hitler suddenly died by a random act or a bomb under the table, this new succession shows a bit more of a "stable", less strongman centric government. This new Nazis government resembled a pre-Hitler German government with a chancellor and president (although the Nazis remained the only legal German party).

This fits in with Hitler's egotistic belief that he was the sole person capable of being the German Furher and leading Germany into its new golden age (thousand year reich); any post Hitler government would theoretically not have been controlled by a single "Furher" because it was believed that Hitler was the only one capable of running such a government. This is actually a bit ironic considering when Hitler first joined the Nazis, he was only trying to get support for the party, he thought of himself as being a vanguard who would pave the way for a more capable leader who would usher in a new golden age for Germany.

So based off this, a new post Hitler Germany would be a single party state with "Reichstag elections" where in the only candidates were Nazis. Presidential elections would happen, but again with only Nazi approved candidates; but they would also most likely not be open to the general public but rather a small group (like a Senate). Hitler actually did have a few Reichstag elections after seizing power (last one was in 1938) and said that there would be further "elections" after the war which again return an entire Reichstag full of nothing but Nazi delegates.

However it is worth noting that Hitler didn't support the idea of a democratic vote for his successors, but rather via a small body like a senate. Since he assumed that would help keep national unit together. He spoke fondly of the electoral system used in the Holy Roman Empire. To quote the man himself:

As regards the Head of the State, should anything happen to me, it would be as unsound to elect my successor by public vote as it would for, say, the Pope to be elected by suffrage among the faithful, or the Doge of Venice by the vote of the whole population of the city. If the mass of the people were invited to take part in such a vote, the whole thing would degenerate into a propaganda battle, and the propaganda for or against any candidate would tear the people asunder. If the choice is left to a small body—a senate, for instance— and marked differences of opinion should arise in it, I don't think it would matter very much, provided that no hint of these differences was allowed to become public.

But once the votes have been cast, then he who receives the majority becomes automatically and forthwith the supreme head of the state. If it is further arranged that the oath of allegiance to the new Head can be administered to the Wehrmacht, the Party and all the appropriate officials within three hours of the result of the election, then maintenance of public law and order can be regarded as assured. I have no illusions, however, that an absolutely outstanding personality will always emerge by this method of selection. But it does at least ensure that the man chosen will be one so much above the average that, as long as the machinery of government is in good order, the State will not be endangered in any way. The old German method of electing the Emperor was an ideal way of forming a government.

However, as this was a statement made during the war, its hard to tell if he meant this in the long run or only as an emergency measure. Either way its a good glimpse into his idea for post-Hitler Germany.

Hitler did intend to serve as an inspiration for future German leaders. In Richard J.Evans Third Reich at War Evans notes that Hitler wanted his musings copied down because he intended future German leaders to use them as guidelines. Things like dealing with the occupied Soviet territories or the inevitable American cold war.

Just to address the idea of "feudal rule" that OP suggested, Hitler actually was not a fan of monarchies, or at least for Germany. He saw royalty as being decadent and in many cases incompetent. So its unlikely he wanted to create an inherent "ruling family" for Nazi Germany. Hitler echoes those sentiments here:

It must therefore be an absolute and fundamental principle of National Socialism that office in neither Gau, State nor Party is hereditary

Also, with regards to the "Thousand Year Reich" it wasn't meant to necessarily be like a literal thousand year state; Arthur Moeller van den Bruck in his book, Das Dritte Reich, argued that that German history could be divided into three periods:

First Reich

Second Reich

Third Reich

This book was influential among the Nazis and was where we get the whole "Reich" terminology from. Van den Bruck argued that the leader of the Third Reich (he was writing in the early 20s before the Third Reich) would be a "German Messiah." So it might be better to look at Nazis saying "thousand year Reich" as the equivalent of "golden age."

And just to note, I'm in no way saying this is what certainly would have happened. As I said we aren't gifted with any sort of sizable knowledge about what Hitler envisioned Germany looking like day to day; this was based off possibilities based on Hitler's actions/beliefs.

Edit:

Added some quotes, changed wording.