Is there a single word for "moving unhinderedly"?

Solution 1:

I suggest powered

He powered through Africa, driving all before him. (example made up by me)

power verb (STRENGTH)

[I usually + adv/prep] to act with great strength or in a forceful way:

Halfway through, she powered into the lead and went on to win the race.

http://dictionary.cambridge.org/dictionary/british/power

Incidentally, I know you have already rejected it but swept is very commonly used in this situation, e.g.

After conquering Syria in 332 BCE, Alexander the Great swept down into Egypt with his army.

Ancient History Encyclopedia

http://www.ancient.eu/alexandria/

Solution 2:

Stormed, swept, blew, breezed, strode, powered, rampaged, stomped, sailed, barged et all, are all fine verbs for the singular act of moving through a place without resistance. Each has its own subtle connotation too, so pick wisely.

To indicate that someone can do so habitually, you may need more than a word.

To have free reign indicates that someone can do with something or go wherever they please within an area. He had free reign of the continent.

To be unimpeded is to encounter no resistance of movement. He travelled the continent unimpeded.

There is a particular meaning to the word check that indicates withholding or limiting something; hence the phrase unchecked power. He moved around the continent unchecked.

One of the beauties of the English language is that for a concept like this, you're not only likely to find dozens of existing examples of florid description, but there is still plenty of untilled fertile ground for further invention. If this concept is central to your story, perhaps coin your own!

Solution 3:

Perhaps storm (as a verb)

to use force to enter a place and take control of it

Macmillan

While this is often used in the context of a battle or a police action, it can have a more figurative meaning, such as

His theatrical performances throughout Europe were amazing. He stormed the continent!

Or you might use the phrase take by storm. Collins offers these two definitions

1. to capture or overrun by a violent assault

2. to overwhelm and enthral

Solution 4:

You might be able to user steamroller or steamroll.

Dictionary.com Unabridged:

verb (used without object)

7. to proceed with implacable force.

He steamrolled across the continent.

Solution 5:

Stride might be suitable. It has a lot of subtly different meanings but the 1st verb definition implies power and arrogance and the idiom "take in stride" implies confidence and success:

stride
verb (used without object), strode, stridden, striding.
1. to walk with long steps, as with vigor, haste, impatience, or arrogance.
Idioms
15. take in stride, to deal with calmly; cope with successfully: She was able to take her sudden rise to fame in stride.

(reference.com)

Therefore, I think it has the desired connotations. The use in the example would be:

He strode through the continent.