The Space-Time Patrol?

The Time Patrol operates at least in the Solar System, not just on Earth, and also has interstellar connections.

(i) Time travel is discovered and the Patrol is founded in 19352 AD when giant combines compete on a galactic scale.
(ii) Everard's training includes how to handle spaceships.
(iii) His trainer was shot off Jupiter in the Martian war of 3890.
(iv) Stane's family was killed in a raid from Venus in the thirtieth century.
(v) Stane steals a time machine from a trader who pretends to be from the Saturnian System, which Stane had visited.
(vi) Unattached Patrol agent Piet Van Sarawak is from twenty fourth century Venus.
(vii) Patrol agent Mendoza works in a hospital on the Moon in 2319.
(viii) The Exaltationists are genetically engineered for adventurousness on the space frontier in the thirty first millennium.
(ix) In 1610, Patrol timecycles attack Exaltationists at Machu Piicchu from an orbiting spaceship.
 (x) Epsilon Korten, director of Jerusalem Base between the birth of David and the fall of Judah, was born in twenty ninth century New Edom on Mars and fought in the Second Asteroid War.
(xi) Another Patrol agent is from the Saturnian System in the thirty second century.
(xii) Time criminals are imprisoned on an exile planet.