Each hero's stats are increased by defeating fugitives. Finding fugitives reliably requires ally Hanzo Hattori. To unlock Hanzo, you first must unlock Fūma Kotarō as playable character by defeating Sanada Yukimura's story, then go through Kotaro's story mode twice - first through his Red Route, then his Blue Route. Hanzo Hattori is found on the stage Cliffs of Mikatagahara.

Fugitives exist on every one of 38 maps in Sengoku BASARA 3. They will randomly respawn in one of three locations once two minutes have passed, or a special requirement is fulfilled, depending on the stage. While their appearance is also random, players can generally get close to a 100% chance of their showing up if they equip a maxed Hanzo Hattori as an ally. He will also mark their location on the map, making it easier to find them.

If you are having trouble spawning fugitives try either killing camp commanders/generals faster or taking it easy. Sometimes you may also just be unlucky.

Fugitives give +50 attack or defense or +1 luck upon finishing map. So killing all fugitives nets +600 attack/defense and 14 luck.

Snegoku Basara 3: Samurai Heroes - Fugitive List
Stage Name Stats Notes
Mount Osore Attack There is a hidden path past west most camp. Look toward the edge facing inwards.
Dragon Race at Oshu Defence Take it slow in the first camp before the 1st gate. Go through the 1st gate at around 1:30-1:50 time
Siege of Hasedo Castle Luck none
Headstrong at Kubota Attack none
Battle of Kannagawa Luck none
Kawanakajima Tundra Luck Open the hidden path by taking the camp and kill the boss within 2 minutes. Grab a horse after taking the North camp
Flooding Ueda Castle Attack Take it slowly within first 2 minutes. Do not kill any bosses or open the eastern gate
Assault on Ueda Castle Defence Defeat Kojoro Katakura within 2 minutes.
Odawara Castle Luck Seize first three camps within two minutes to lower the drawbridge. Take the sneak attack route through it
Kaerikumo Castle Luck Go fast and try to lower all bridges within first 2 minutes.
Battle of Tetorigawa Defence Go slowly do not kill any bosses within 2 minutes.
Cliffs of Mikatagahara Luck Force Tadakatsu Honda to attack with lightning within 2 minutes.
The Last Stand Defence none
Sekigahara: Betrayal Attack none
Sekigahara: Schemes Attack none
Sekigahara: Zero Hour Attack none
Sekigahara: Assembly Defence none
Sekigahara: Ultimate Defence Wait 2 minutes before attacking anything.
Sekigahara: Aftermath Defence none
Sekigahara: Intrusion Luck Do not take any camp within 2:10 minutes.
Kanegasaki Nightmare Attack On the cliff path, take one of the two launchers.
Siege of Kanegasaki Defence none
Incident at Honno-ji Luck none
Osaka: Winter Attack none
Retaking Osaka Castle Defence none
Osaka: Tiger Summer Attack none
Osaka: Dragon Summer Defence none
Saika Stronghold Luck none
Siege of Karasu Castle Luck Take Tenkai's path and the fugitive will spawn within 2 minutes.
Gassantoda Castle Luck none
Battle of Itsukushima Attack none
Nakatomigawa Fortress Luck Do not destroy the rising sun within 2 minutes.
Battle of Noshima Attack Summon the allies boat by beating all the squad leaders in the first camp then make Tsuruhime retreat before she summon another allies boat.
Pirates of Noshima Defence none
Mines of Ishigakibaru Luck Found in the Engineer spawnpoint in the 2nd Engineer camp.
Hetsugigawa: Memories Attack none
Hetsugigawa: Heroes Defence none
Ambush at Mimikawa Luck none