First of all the MudrA in the images is not ApAna MudrA at all. It is something else.
ApAna MudrA is formed when the middle, ring fingers and the thumb are conjoined at their tips.
There are the 5 principle Vayus viz - PrAna, ApAna, VyAna, UdAna and SamAna and we have 5 corresponding MudrAs too for each one of them. These MudrAs are used during doing Naivedyam to Gods. They are part of the GrAsa MudrAs. So, descriptions of how to form these Mudras are found in several Tantras.
PrAnApAna samAnAshcodAna vyAnau cha vayovah |
jneyo prAnAh pancha samiritAh ||
PrAnamudrA samAkhyAtA prAne
samakhyAtA prAne havankarmani |
madhyamAngushthaistribhirekikritam yadi ||
tathAhanAmika madhyame |
Kanishthena samAyuktA niyuktA vyAnahomake
Nishkanishthena yA mudrA sodAna havane smritA |
samskritA mudrA samAnAhuti karmAni ||
The five VAyus are - PrAna, apAna, SamAna, UdAna and VyAna.Among these
five, know SamAna to be the fith one. When offering oblations in PrAna
VAyu, the PrAna MudrA is prescribed. When index finger, middle finger
and the thumb are conjoined then PrAnamudrA is formed. When the ring
finger, middle finger and the thumb are likewise joined then the ApAna
MudrA is formed and in ApAna-oblations this MudrA is to be used ......
ShAktAnanda Tarangini 14-Verses Collection 22.
These MudrAs are also described in the book Nityotsava which quotes from another Tantra and there the description of the ApAna MudrA is different from the one given above. But, at the same time it does not match with the images found in the question too.
SwAmi RAmadev's book Yoga SAdhanA also has pictorial descriptions of this and other MudrAs. See the pic given below (the one in the middle is the ApAna MudrA):
From, Upanishads, Tantras we get to know that there are certain working areas for each of these VAyus. For, ApAna Vayu, this area is Guhya or anus. So, it is basically related to excretion.
According to the book Yoga SAdhanA, the following are the benefits of performing ApAna MudrA:
By performing this MudrA, the toxins are removed from the body. It
helps in curing constipation, gaseous troubles, diabetes, kidney
problems and tooth problems. It is generally very good for all bowel
problems. It also helps in heart problems and creates sweating.
What RajnikAnt is doing is probably showing a sign of assurance or victory to his followers. And, what Sri RAmkrishna was doing will be better known only to him because, as is said,he was in a state of SamAdhi when that pic was taken. Also, note that his left hand is in some other MudrA. But, in ApAna MudrA both hands should be in the same posture.
So, the conclusions here are:
- The MudrA in the images is not ApAna MudrA.
- Besides being used in rituals, it has immense health benefits as well but the spiritual benefits are seemingly none or not significant.