Board Thread:The Panopticon/@comment-45314928-20200525010944/@comment-6032121-20200525101616

See my message above, showing that the River Song problem is a red herring — we've already got valid sources where she has other adventures with the Tenth Doctor.

But at any rate, I think any sort of distinction based on whether the Doctor invited the maybe-companions to travel with them is fundamentally, even ghoulishly flawed. The archetypal companions, Ian Chesterton and Barbara Wright, were famously not invited to travel with the Doctor, but rather kidnapped; they and the Doctor spent the rest of the era waiting for the Ship to land in the 1960's again so they could go home, for all that they ended up striking some sort of friendship with the Doctor and Susan in the meantime. There are also several other Classic and EU companions who stowed away aboard the TARDIS or were otherwise accidentally taken on-board, and who only continued travelling with the Doctor because they had nowhere else to go.