A strict interpretation of the rules requires you to question the timing of a free action. For instance, say I'm a rageblood barbarian and I charge a foe, and then I crit and drop that foe to 0 hp using howling strike. Strict interpretation of the rules says that I do indeed trigger the Rampage class feature and swift charge power. However, I can't actually use them because it's still my turn and I can't take any further actions on my turn. I then enter my End of Turn phase, and I do what I need to do in that phase. Then someone else's turn begins. (As Saeviomagy points out, this technically means you can't even speak until the next turn begins.)
That's how I'd have to rule it in a competitive tournament game, such as the Ultimate Delve, where player knowledge and tactics really matter.
In my home game, as well as a noncompetitive RPGA game, I'd allow the player to choose to use both Rampage and swift charge in this case. From what I've seen, everyone at the table has more fun when these barbarian mechanical elements (and similar ones from other classes) are brought into play. The triggers happen rarely enough that restricting the use of these capabilities seems a lot like restricting the fun.
I'm a little lax with the rules when doing so results in more fun. That laxity sometimes benefits the monsters, so it all comes out fairly.
To some folks, however, it's more fun to acknowledge the limits of the rules and play in a tactically minded way. My players aren't among such tacticians, so I'm lax. If I had a lot of tacticians in my group, I might change up my style and be stricter with rules interpretations to reward really careful tactical play.
For any home game, it's a judgment call. Consistency is key.