What puts together composites according to Buddhism?
The word 'sankhara' has many different meanings. It can mean:


"conditioned thing' (something made from causes &amp; conditions; AN 3.134; Dhp 277; SN 22.90)
"conditioner' (a condition that conditions/creates another conditioned thing; SN 41.6; MN 44).
"conditioning" (the process of conditioning; SN 22.79)
aggregate of mental formations (sankhara khandha; SN 22.79; SN 22.48)
mental conditioning (i.e., mental proliferation, psychobabble or papanca; Dhp 203; SN 38.14).


Therefore what puts together a conditioned thing (sankhara; sankhata) is the conditioning (saṅkharonti) of other conditioned things (sankhara). 

In the context of mental formations (sankhara khandha), SN 22.79 says:


  Kiñca, bhikkhave, saṅkhāre [plural] vadetha?
  
  And why do you call them 'fabrications'?
  
  Saṅkhatamabhisaṅkharontīti kho, bhikkhave, tasmā ‘saṅkhārā’ti vuccati.
  
  Because they fabricate [saṅkharontī; verb] fabricated things
  [saṅkhatam], thus they are called 'fabrications'[saṅkhārā]
  
  Kiñca saṅkhatamabhisaṅkharonti?
  
  What do they fabricate [saṅkharonti] as fabricated thing [saṅkhatam]? 
  
  Rūpaṃ rūpattāya saṅkhatamabhisaṅkharonti, vedanaṃ vedanattāya
  saṅkhatamabhisaṅkharonti, saññaṃ saññattāya saṅkhatamabhisaṅkharonti,
  saṅkhāre saṅkhārattāya saṅkhatamabhisaṅkharonti, viññāṇaṃ viññāṇattāya
  saṅkhatamabhisaṅkharonti.
  
  For the sake of form-ness, they fabricate form as a fabricated thing.
  For the sake of feeling-ness, they fabricate feeling as a fabricated
  thing. For the sake of perception-hood... For the sake of
  fabrication-hood... For the sake of consciousness-hood, they fabricate
  consciousness as a fabricated thing.
  
  Saṅkhatamabhisaṅkharontīti kho, bhikkhave, tasmā ‘saṅkhārā’ti vuccati.
  
  Because they fabricate fabricated things, they are called
  fabrications.


In the context of intentional kamma (action), AN 3.23 says:


  A certain person makes/generates/puts together
  (abhisaṅkharoti) hurtful (sabyābajjhaṃ) body
  choices/intentions/activities (kaya-sankhara), speech
  choices/intentions/activities (vaci-sankhara) and mental
  choices/intentions/activities (mano-sankhara).
  
  Saṅkhāra Sutta