These are imperial legions... requests of this nature went from a provincial governor/legatus to the emperor and/or his court.
It depends on what they needed, and where they were. Under most circumstances, as already suggested, auxilia would be raised locally, but there are cases of major movements... such as the Sarmations to Britain under Aurelius. A legionary cohort could not be raised without the order of the emperor. Vexillations (detachments) of legions were commonly transfered in various places throughout the empire depending on need.
Vegetius provides an excellent starting point into many of these issues.
Portus Ritupiae (near modern Richborough, near Dover) would still have been the major British port circa AD 60. The likely transport route from the continent to Britain was probably Gesoriacum (modern Boulogne, France) to Ritupiae. (This is the route we know that Plautius used for the invasion in AD 43)
That all depends on who, what, why and how... I'm sure some liberties can be taken with this.
This site Roman Britain.org is an invaluable resource on Roman Britain. You may have to do a bit of research, but you should be able to find some actual auxilia and or legionary units that may have been arriving about the time period you are looking for.