Continuity mistake: During the crashes by the border control the road keeps swapping from sunny to shadowy back and forth in less than 3 seconds.
Continuity mistake: When Indy jumps on Kazim's boat, Elsa's boat is inches away from Kazim's, but a shot later she has managed to appear somewhere else with no boats around.
Continuity mistake: When Marcus watches the carbon copy of the shield, his ice-bag is on the table, but when the angle changes he has it next to his head and seconds later he puts it on the table. Also, the ashtray moves backwards between shots.
Continuity mistake: On the tank, Indy is punched and the gun falls inside. His hat also falls off, but a shot later it's back on, perfectly tucked in.
Continuity mistake: After Dr. Jones fires at the jeep two soldiers fly towards the tank, but when the angle changes they're falling away from the tank.
Continuity mistake: Inside the temple, when Indy is at the cliff, in the close-ups the rocks next to him are round shaped, but in the wide angle they are straight and stripped.
Continuity mistake: When about to walk on air in the temple, Indy says, "No one could jump this". In the close-ups his hands are spread apart, but in the wide angle they are grabbing the diary.
Continuity mistake: By the cliff, Dr. Jones hugs Indy, whose jacket is down at waist level. When Dr. Jones leaves, Indy collapses on his knees and now the jacket is perfectly placed over his shoulders.
Continuity mistake: The German Nazi suddenly disappears from the tank when it crashes on the ground (as is replaced by an obvious scale model).
Continuity mistake: When Indy leaves the steering wheel to stop the baddie on his motorboat, he passes by a metal turret twice in less than half a second.
Continuity mistake: During the fight on the tank, Indy punches a German who falls on the wheel and whose hat falls near the turret. When the angle changes the hat is in front of him, but when he falls off the tank the hat is gone.
Continuity mistake: The plane explodes at the exit of the tunnel. A frame later the exit is clean and the small remains are scattered around.
Continuity mistake: When Indy and Elsa encounter the rats, hundreds of them block the way, making Indy lift up his legs. However, a frame later, when Elsa starts to walk, those rats have disappeared.
Continuity mistake: After escaping from the turning blades, Indy's hat swaps from clean to filled with spiderwebs. Though there are 4 seconds in between, he didn't move from the previous spot, nor were there any webs around.
Continuity mistake: Right before the castle is on fire, there is a chair very close to the fireplace. When the place starts to burn, father and son take shelter under the fireplace thanks to the fact that the chair has magically moved several meters away.
Continuity mistake: While in the train station in Iskenderun, Marcus passes by the same spot and the same people twice (watch the man with the hen on the left).
Continuity mistake: When Donovan turns into a living corpse the badge on his lapel disappears.
Continuity mistake: When Donovan drinks from the grail his left hand holds the base with extended fingers. A frame later they are bent.
Continuity mistake: In the catacombs, Indy finds the tomb and blows dust away from the shield, cleaning the sides of the engraved cross, but leaving the rest very dusty. A frame later, the whole cross is spotless.
Continuity mistake: While the propeller chops the boat, Indy grabs Kazim and lifts him up saying, "This is your last chance." But from the opposite angle Kazim is still seated.
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 ★