And do these practices continue today? Are exceptions made for children, the old and so forth?
This article from a sri vaiShNava thirumaligai (religious center) says:
The AchAryA embosses the impressions of Conch (Shankam) and Discus (Sudarshana chakram) on the left and right arms of the disciple, respectively. The emblem of”Sudarshana ChakrA (Discus)”is heated in the agni and is embossed on the right hand, by chanting a hymn, which translates into “Oh! Sudarshana MahAJwAlA (Highly resplendent), as lusturous as crores of Sun, please show me, an ignorant one, the path of Lord Vishnu”. Then, the emblem of”PAnchajanyam (Conch)”is heated in the fire and is embossed on the left hand, by chanting a hymn, which translates into “Oh! PAnchajanyA! You are capable of driving away the sins of individuals through your sound. Please protect me, a sinner, from this samsArA”
Another article from a website connected with a sri vaiShNava mutt says:
heated impression of Sanka and chakra on our shoulders. This identifies that we are property of emperumAn - just like a vessel is marked with the symbols of the owner, we are marked with the symbols of emperumAn.
[the branding marks] must be clearly visible. This mudhrA is embossed on the shoulders by heating the sangu (conch) and chakram (disc) with specific manthrams by the AchAryan. While it may appear like a boil/wound initially, eventually it will heal and the mark remains on the shoulders forever. This is the first samskAram in samAsrayaNam or pancha samskAram - the process of initiating oneself into srIvaishNava sampradhAyam. It is essential that the marks remain clearly visible on the shoulders for srIvaishNavas. Not having visible marks is an obstacle. Translator's note: In many AchArya pItams, it is seen that, the conch and disc are heated moderately and placed gently on the shoulders of the sishyas - this ends up in not having very visible marks on the shoulders. As this practice is seen in practice in many AchArya pItams, it is considered as sistAchAram (that which is done by our elders) in this case. There is another way of accepting thApa samskAram - which is using flowers - this is called pushpa samAsrayaNam/samskAram. This is not seen in our sampradhAyam at present. This practice is pursued by asishtas (people other than our pUrvAchAryas) and thus is not to be pursued by srIvaishNavas. thApa samskAram using heated conch and disc is the proper way of getting initiated. Not doing that is an obstacle.
Unfortunately I have not been able to find references specifically to branding in sri vaiShNava granthas, divya prabandham or in smRRiti literature. I would be very surprised if it is unmentioned here, so I would appreciate additional quotes. Neither have I been able to find relevant information on the dvaita sampradAya's way of making these marks: this unanswered question is relevant What is the initiation process for the Dvaita sampradaya? In what aspects does it differ from the Panchasamskara done by Srivaishnavas? .
GauDIya vaiShNava do not practice this as far as I know. Their text prameyaratnAvalI (8.6) says:
tapo... tapta-cakradi- mudra-dharanam ucyate tenaiva hari-namadi- mudra capy upalaksyate. hari-namaksarair gatram, ankayec candanadina...
the word austerity means to accept the branded marks of Lord Visnu (the cakra, lotus, conch and mace) and also the marks of His holy names. One who marks his body with the letters of Lord Hari’s holy names drawn in sandalwood paste or other similar substances...
A note is added to this text, saying:
"Srila Bhaktisiddhanta Sarasvati Thakura comments that this difficult austerity is not actually needed. The previous great devotees have set the example of marking the body with the Lord’s holy names and symbols drawn in sandalwood paste. This is quite sufficient.". This suggests that Gaudiyas today do not brand themselves with symbols.
Bhaktivinoda ThAkura is a purvAchArya of all gaudiyas today, so his statements have authority on how they practice initiation. He wrote a whole article about how initiation works for Gaudiyas, which mentions that:
In the Śrī sampradāya of Rāmānuja, tāpa is given by branding the body with the symbols of conch and disc, but Śrī Caitanyadeva has instructed that we mark the body with hari-nāma using sandal paste etc. instead of brands. This rule is a blessing for the souls of Kali-yuga.
sanatkumAra saMhitA is another text that is closely related to the Gaudiyas, but also important for other similar vaiShNava such as nimbArkas. Verse 100 says:
chandanena mRRidA vApi vilikhet bahu-mUlayoH, vAma-dakShiNayor sha~Nkha-chakre yathA kramam.
using sandal paste or clay, the spiritual master should draw a conchshell and cakra on the left and right shoulders in this order.
When I was initiated into the nimbArka sampradAya by one of my family, it followed the order here and I was only given a temporary mark in sandal paste, but this that some nimbArkas make marks with branding, and some with clay or sandal.