The story Lupin III vs. Detective Conan revolves around a
mystery (that's why Conan's there) and a heist (that's why Lupin's
there!), which involves a mother and son, royalty, found dead, an
accidental death to the mother and a supposed suicide of the son.
Now all that's left is the princess, who does not want to be queen,
is hated by the people, and wants her beloved family back. Jigen
(Lupin's friend) ends up being assigned a job to protect the
princess (he's training military also!), the princess goes to an
opening ceremony of a hotel in Japan dedicated to her mother, she
meets Ran on the streets while running away (you should know her
and Ran look exactly alike), the two switch roles, a woman named
Fujiko (you have to know Lupin III to know Fujiko) takes the
princess, now disguised as Ran, on an adventurous trip, and chaos
befalls Ran, and Conan meets the famous notorious thief Lupin III
(and calls Jigen "papa". Heehee.)