Continuity mistake: When Indy is trying to hold Elsa and keep her from falling and she's trying to reach the Grail, you can see her hand coming out of her glove. In the first shot of her hand coming out, it comes almost all the way out. In the next two shots, her hand is about halfway out, then it slips out and she falls. (01:57:00)
Continuity mistake: In the temple in the Canyon of the Crescent Moon, Indiana Jones rushes to his father's aid with the Holy Grail filled with water. He is shown pouring every last drop onto his father's wound by turning the Grail upside down, yet another camera angle shows his father clutching the grail to his chest, and there's still a bit of water in it.
Continuity mistake: When Indy is stepping on the letters, he steps on the "O" and knocks the "P" loose. In the next shot, the "P" is intact.
Continuity mistake: In the motorcycle chase scene after the escape from Brunwald castle, Indy's hair is noticeably longer than it was in the castle scenes. It returns to the shorter length when they stop at the crossroads.
Continuity mistake: In the beginning of the boat chase two bullets hit the windscreen of Indy's boat. In the next shot the location of the holes has changed.
Continuity mistake: As Indy swings through the window to rescue his father, he gets hit on the head by a vase. His hat is all crumpled and tipped quite far back, then in the very next shot his hat is neat and level.
Continuity mistake: When Indiana is showing Marcus the Grail Diary in his fathers home, he closes the book 4 times in a row without opening it - first 3 times inside the room, and then once on exit.
Continuity mistake: When Indy and Elsa escape underwater, and surface from the tunnel, Indy's suit should be drenched, but it only has water stains. Then during the speedboat chase, their clothes are completely dry.
Continuity mistake: In the boat chase, when the bad guys are using machine guns, in one shot we see Indy's boat in the middle of the canal. In the next shot (close-up) they are very close to a large pile of dirt (or wood chips) and in the next (wide) shot they are in the middle of the canal again.
Continuity mistake: When Indy comes face-to-face with Hitler he hands over Henry's diary with its black elastic around the back cover, but in the closeup as the signature is being signed the elastic is around the front cover, then it's around the back cover again when Indy takes the diary back.
Continuity mistake: The sky colour/condition changes between the shots of Indy and his father in line to get on board the Zeppelin and the scene just before. In one, it is cloudy, and in the other it is a clear blue.
Continuity mistake: At the very end there is a large near circular blood stain on the left sleeve of Sean Connery's shirt. When they ride off, the stain has disappeared completely.
Continuity mistake: When the military caravan with Brody is driving through the canyon towards the Joneses and Sallah, the sun is behind Brody (discernable by the shade). But when Indy is looking down on them - from the front - the sun is also behind him.
Continuity mistake: When Old Dr. Jones is holding Young Dr. Jones while he is reaching for the grail, watch Indy's left hand. In the shot from above, it is by his side, in the shot from below, he is holding the rock. It keeps switching from his side to the rock.
Continuity mistake: Indy falls off the side of the tank and gets his shoulder strap caught on the gun - this should not be possible because his shoulder strap is underneath his jacket, as can be seen when he gets back up onto the tank.
Continuity mistake: When the Nazi is strangling Indy with the chain, Indy looks down into the porthole of the tank and yells, "Dad!" Before he throws the gun down into the tank Indy's hat comes loose, but when the next shot is back on Indy, and he yells, "Dad! Dad, get out!" now his hat is back to normal.
Continuity mistake: After Sr. says "Let it go" as Indy is trying to get the grail, Indy is swinging and not being supported by the wall. The next camera angle shows him grasping the wall near his head. Switched back to Sr.'s view, he is again not holding the wall.
Continuity mistake: When Indiana is drinking champagne with the businessman in the fancy apartment he holds his glass by the stem in one shot and then by the bowl in the next shot. As the conversation continues the glass goes up and down in Indiana's hand like a yo-yo.
Continuity mistake: When Marcus is taken to the temple by the Germans, the knot on his tie differs between a far shot and a close one.
Continuity mistake: When Indy and Vogel are fighting atop the tank we see the soles of Vogel's boots and there is a red hexagon dot under each heel, but while Indy is hanging off the tank's side mount gun the soles of Vogel's boots are different.
Chosen answer: They're not, no. An individual doesn't become immortal after one drink - it requires them to drink regularly in order to remain alive. So neither Jones has been rendered immortal, merely healed of any wounds that they might have. But your surmise is basically correct - as the Grail cannot leave the shrine, any individual wishing to use it to prolong their life must stay there if they wish to enjoy its effects.
Tailkinker ★
Also, in Indiana Jones and the Kingdom of Crystal Skull, Henry Jones, Sr has passed away before the start of the story and therefore was not immortal.
raywest ★