they sit in a crib, crawl around, stand-sit down, bla-bla, sometimes mantra cry or calls for parent and then just lay down and fall asleep?
Although I knew if mine was sitting up or standing up (he couldn't stand at 5 months though so I mean older) then it meant there would also be call-backs, him either shouting for me or crying which both require a response. These would be times when there was something disturbing him, teething, illness, developmental leaps or if our routine needed a tweak and he was perhaps UT or OT due to the routine being in a transitional state. All of these things would require some additional help at BT despite him being an independent sleeper since about 8-10 wks.
When he was up and walking we had some long BTs with him running up and down the cot.
Independent doesn't mean never ever needing any help ever again. It does mean that when the routine is settled and appropriate, when there is not too much pain from illness/teething then the normal way of things is to put LO in the cot say goodnight and leave the room.
and what time does it take with your children to fall asleep such way?
We had a wind down of just a few mins. Upstairs to bathroom, change nappy, into bedroom (already dark), one song (or half a song if he cued that he wanted down faster), put into cot, say "have a nice sleep, call if you need me" and leave the room. He didn't like to fall to sleep with me in the room but he would nod off within either a few seconds or 2-3 mins once I'd left the room. He used a muslin square as a lovey from around 6 months and would rub this on his face and play with it in his hands. If he mantra cried fro longer than 5 mins I knew something was wrong and he would need additional help. Although it is quite normal for a LO to take 20 mins to fall to sleep so a mantra for 20 is fine.
are you/anyone sitting in the room while the baby is trying to fall asleep?
Only during those tricky periods mentioned above.