Succession doesn't need to pass from grand-father to grandson.
Emperor Karl I of Austria was the grand nephew of Emperor Franz Joseph.
While Rudolf was styled as Crown-Prince during his father's reign, AFAIK Karl was just the
presumptive heir (just like his father Franz-Ferdinand had been until his assassination in Serajevo)
I wanted a clarification of this designation (notwithstanding the fact that the title is not necessarily used in all kingdoms): has it traditionally been reserved only for an heir-apparent who is a son of a king? What if he is a grandson of a king, whose father (the previous crown prince) has predeceased him?
If a prince is born second and direct in the line of succession, and his father dies, does he become the new crown prince? I believe that the title Hereditary Prince is used in some kingdoms, to denote a direct heir is not first in line to the throne ...
Of course, it is also used to denote the heir-apparent to a principality (e.g. Monaco and Liechtenstein). It goes without saying that the title Crown Prince is never used in those places -- or even in grand duchies (e.g. Luxembourg).
Can "Crown Prince" actually be used for an heir-presumptive who is first in line to the throne -- with the understanding that the said person can lose that designation?
I think it might be different from country to country. I will give an answer how it is in Sweden.
When Oscar I died in 1859, his elder son, Crown Prince Carl, became King. Carl XV's younger brother Oscar was Heir Presumtive, but he was never Crown Prince, nor did he hold a special title showing that he was first in line of succession.
When Gustaf V died in 1950, his eldest son, Crown Prince Gustaf Adolf, acceded to the throne as King Gustaf VI Adolf. The new King's eldest son, Gustaf Adolf, had died in 1947, so the King's grandson Carl Gustaf became Crown Prince.
In 1973, King Gustaf VI Adolf died, and his grandson, the Crown Prince, acceded to the throne as King Carl XVI Gustaf. The new King had no son. Therefore, his uncle, Prince Bertil, became Heir Presumtive, but with no special title to show that he was first in line of succession. In 1979 the King became father to a son, who from birth was Crown Prince Carl Philip. In 1980 a change of the Act of Succession allowed for the eldest child regardless of gender, to inherit the throne. Thus, the King's daughter Victoria, who was born in 1977, became Crown Princess.