Our evidence on this question is limited, and mainly comes from two sources: inscriptions on gravestones and letters, often letters of recommendation.

These both indicate that Roman soldiers could and did serve in multiple different legions - or at least those of above average rank did. One famous example of a Roman soldier's tombstone is that of Petronius Fortunatus, found in North Africa.

It describes a remarkable career. He enlisted in Legio I Italica in the late 1st century A.D., and held the posts of librarius, tessarius, optio and signifer over a four year period. (All junior officer ranks.) Then he was promoted to the rank of centurion by vote of his comrades. (A holdover of Republican times, but apparently still sometimes done in Imperial times.)

He then spent the next 46! years as centurion serving in Legio VI Ferrata, I Minervia, X Gemina, II Augusta, III Augusta, XXX Ulpia, III Cyrenaica, XV Appollinaris, II Parthica, I Adiutrix. This would have taken him all over the Roman Empire, since these legions were stationed in Britain, Africa, Syria, Germany and in the Balkans. (Where he started and ended his career.) He died at age 80.

It also mentions he was decorated with the mural crown, awarded for the first man to cross the wall of an enemy fort of town. Also had other decorations including torques and phalerae. Finally, it says he had a son who also served as centurion, which probably means he was directly commissioned. (This was the way most centurios got their posts.)

Such grave inscriptions are an invaluable source of information, because they tell us what these people (or their family) thought was most important to tell posterity. It shows what they valued and took pride in.

The downside is that they come with an inherent bias: only the well-off and succesful could leave extensive inscriptions such as this (though by no means only the very rich.) and they are only likely to note remarkable or unusual things. They also offer little in the way of details: the above example is one of the more elaborate ones we've got, but it only lists the simple facts and offers no reasons or explanations.

Letters are more rarely preserved than gravestones, but may offer more revealing information because they are written to address specific concerns and are not written with hindsight. There are many letters of recommendation preserved, which are often very simple and formulaic, but seem to have been one of the main tools for advancing one's career. Patronage and personal connections were critical in the Roman world.

Personal letters are even more revealing, but also rarer. Still, we do have some. A great example is the letters from the archive of Gaius Iulius Apollinarius (and his father), a wealthy Graeco-Egyptian from the 1st century A.D. (His father Latinised his name on joining the army.) Because we found part of the entire family archive, we can reconstruct his life and career in much greater detail than for just about any other soldier.

Apollinarius joined the locally stationed legio III Cyrenaica, but moved with his legion to the city of Petra when Trajan started his annexation of Arabia. He writes how he has asked with the local governor to join his staff, in part so he won't have to do the hard labour of a common soldier. Though his fails, he gets another improved position. Later he writes about being transferred to a cohort in Bostra.

Unlike Fortunatus, Apollinarius seems to have served in the same legion his whole career, but did transfer between cohorts and did serve in multiple ranks and multiple places.

Again, we see a pitfall with such sources: it's likely the wealthier and better educated men writing these letters, and it's also very random which ones happened to be perserved. This makes it very hard to quantify how common this was. We'd expect that many soldiers, especially conscripts from the lower classes, would not have such opportunities. As another soldier writes in a papyrus:

‘nothing happens without money, and letters of recommendation have no value unless you help yourself’ - P. Mich. VIII 468, 38-41

Still, the reasons for seeking transfers are clear enough: Roman soldiers tried to advance their careers, and did so through making personal connections who might help them obtain better positions elsewhere, with more pay, more prestige, or less onerous duties. This fit in the general system of Roman patronage and clientele.