Against the river
The rules for Flowing Water can be found on the core rulebook (page 432):
Flowing Water: Large, placid rivers move at only a few miles per hour, so they function as still water for most purposes. But some rivers and streams are swifter; anything floating in them moves downstream at a speed of 10 to 40 feet per round. The fastest rapids send swimmers bobbing downstream at 60 to 90 feet per round. Fast rivers are always at least rough water (Swim DC 15), and whitewater rapids are stormy water (Swim DC 20). If a character is in moving water, move her downstream the indicated distance at the end of her turn. A character trying to maintain her position relative to the riverbank can spend some or all of her turn swimming upstream.
Swept Away: Characters swept away by a river moving 60 feet per round or faster must make DC 20 Swim checks every round to avoid going under. If a character gets a check result of 5 or more over the minimum necessary, she arrests her motion by catching a rock, tree limb, or bottom snag—she is no longer being carried along by the flow of the water. Escaping the rapids by reaching the bank requires three DC 20 Swim checks in a row. Characters arrested by a rock, limb, or snag can’t escape under their own power unless they strike out into the water and attempt to swim their way clear. Other characters can rescue them as if they were trapped in quicksand (described in Marsh Terrain).
Do note that the surface of water has generally the same speed, while undertowns (faster movement under the surface) and riptides (ripples caused by wave movement near the bank) will not affect the surface currents.
If they are all on rafts or logs going downwards the river, they will follow the river current's speed, regardless of their skill checks. The rules for this can be found on the vehicle combat rules from Ultimate Combat:
Current: From canoes and large ships to winged gliders, vehicles propelled by currents typically manipulate an already existing power source within or outside of nature—an air current, a water current, or more exotic currents, like conduits of magical energy. Usually, manipulating a current-propelled vehicle requires a skill like Fly, Knowledge (nature), Profession (sailor), Survival, or even Acrobatics or Knowledge (arcana), depending on the nature or makeup of the vehicle and the current the vehicle is manipulating.
Water Current: Vehicles that only rely on currents of water for their propulsion are somewhat limited. These vehicles can only move in the direction and at the speed of a current unless they also employ some other means of propulsion or manipulation, and thus often have an additional form of propulsion, such as muscle in the case of a canoe, and wind in the case of a galley. A current-driven ship such as a river barge with a crew of two or more creatures requires either a Profession (sailor) or Knowledge (nature) check for the driving check, as ships require precision, discipline, and knowledge of the natural world. Smaller water-current vehicles, like canoes, use the Survival skill as the drive skill, as reading the terrain is a very important aspect of maintaining control over those types of vehicles.
If it moves with the current, a water-current vehicle's maximum speed depends on the speed of the current (often as high as 120 feet). The acceleration of a water-current vehicle is 30 feet.
A Raft vehicle can move following the river current's speed with no acceleration from the vehicle, or have muscles properling it forward to increase the acceleration up to 30 feet per turn. So, if the river currents are at 60 ft/turn, you can propel it further to 90 ft/turn going with the current, or attempt to go against the current and reduce the vehcile speed to 30 ft/turn.
Futher reading, we see that the driver of a vehicle can propel the vehicle forward or backwards by using vehicle specific actions (Accelerate, Decelerate, Keep Going, Turn, etc), or simply leave the vehicle Uncontrolled, which will make it move foward with a decreasing speed of 10 ft per turn. So in 3 turns left uncontrolled, a Raft vehicle would have acceleration 0 and move according to the river current's speed.
Attempting to jump from a raft to another can be done with Acrobatics checks. Since the terrain slighly unsteady the DC increases by +2 for every check attempted.
If one raft is adjacent to another, the jumping distance is 5 feet (DC 5). However, since you do not have a running distance, the DC is doubled (DC 10). So jumping from your raft to another that is adjacent to yours should be a DC 12 Acrobatics check.
If the rafts are further than 5 feet away, the DC increases considerably (10 feet = DC 22, 15 feet = DC 32, etc). Additionally, the character who fails his check by 5 or more falls into the river, but if he fails for 4 or less, he can make a DC 20 Reflex check to grab hold on the destination raft.
Crashing your raft against rocks or another raft could be a problem aswell, on top of taking damage, the vehicle takes -2 to AC, the DC to control the vehicle also increases by +2. If it happens to be destroyed, then it sinks and all characters will have to make the proper Swim checks against the current.
Sudden Stops: When a vehicle comes to a sudden stop—its movement is reduced to 0 in some way other than the driver using a drive action to slow the vehicle, both creatures and items on the vehicle are violently pushed toward the vehicle's forward facing a number of squares equal to 1/2 the vehicle's current speed before it came to the sudden stop. This movement does not provoke attacks of opportunity. At the end of this movement, creatures and objects take 1d6 points of damage, and creatures must succeed at a DC 20 Reflex saving throw or be knocked prone. If the movement pushes creatures or objects into solid objects, that creature or object takes an additional 1d6 points of damage for each 5-foot square the push was reduced by the solid object.
For instance, if a vehicle with a movement of 60 feet makes a sudden stop due to hitting a brick wall, its driver is thrown 30 feet toward the brick wall. If the brick wall was only 5 feet away from the driver at the point of impact, the driver moves forward 5 feet, hits the wall, and takes 5d6 points of damage. She then takes the original 1d6 points of damage, after which she makes a Reflex saving throw to see if she falls prone for the sudden stop.
Additional rules can be found on the People of the River Player Companion book (page 24):
Catching a Creature Floating Downstream: As long as you are on a riverbank, water vessel, or overhang such as a dock or tree branch, you can attempt to catch a creature floating downstream as long as its path takes it through a space adjacent to yours. In order to successfully grab the creature, you must succeed at a Strength check (DC + 15 + 1 for every 10 feet per round the creature is traveling downstream). For example, catching a creature traveling 60 feet per round downstream would require a successful DC 21 Strength check. If the floating creature is helpless or unconscious, the DC increases by 10. If you are standing on an uneven or unstable surface, the Strength DC increases according to the Acrobatics Modifiers table on page 89 of the Core Rulebook. If you fail your Strength check by 4 or less, you simply fail to grab the creature and it continues downstream; failure by 5 or more means you are potentially dragged into the water as well, and must succeed at a Reflex save (DC = Strength DC above + 5) to avoid the same fate as the creature you tried to help.
You can also use a long, sturdy object such as a pole, loose tree branch, or reach weapon to pull someone out of the water from up to 10 feet away, though in this case you merely brace yourself as best you can and the creature drifting downstream must grab the object, requiring the floating creature succeed at a Reflex save (DC = Strength DC above). You don’t risk being pulled into the water when using an object to catch a creature drifting downstream.
The book also brings another rule related to fighting near a river, that could be useful to your encounter:
Swinging from Vines: As a full-round action, you can swing using a rope, vine, or similar aid within reach toward an opponent and make a single melee attack. You must move at least 20 feet (4 squares) and you must start on elevation that is equal or higher than that of your opponent. Your movement provokes attacks of opportunity as normal. This action is otherwise treated as a charge attack.
Finally, the Cerulean Seas is a third-party campaign setting that focus on underwater campaigns, and it has several rules for both underwater combat, fighting currents and underwater hazards. It's worth checking it out if you are interested on that kind of enviroment play.