I found a lot of translations using “pour la forme”.

Even though it is (just a bit) familiar, it really means that, you did something, but you do not really believe in it and just did it “by convention” or because you had to / were forced to.

A

“To go through the motions” means to do something in such a way that every step in the sequence of operations or tasks involved is addressed or let’s say most of them, the most important ones, but each step is taken care of willy nilly, without a real intent to do a good job of it; in other words work of mediocre quality could very well be the result. The French equivalent appears difficult to determine; it could range from “bâcler” as one extreme to “faire qqc superficiellement” (reverso).

• bâcler
• faire qqc sans gout
• faire qqc superficiellement

B

There is another aspect to the meaning of this locution and it is not taken into account by the above; it is brought to our attention by user 3177’s answer. This added acception makes abstraction of any possible bad quality of the work undertaken or of how well is carried out the activity that’s gone through in such a way that is justified saying that the person is going through the motions. The translation proposed by user 3177 is then appropriate (machinalement, machinal). Numerous expressions have been gathered below as all might supply a good possibility of translation; however they must be adapted, slightly modified according to the context.

• faire machinalement
• faire qqc sans y croire
• faire qqc sans s’y intéresser
• faire qqc par routine
• faire qqc parce qu’il faut le faire (un peu familier)
• faire qqc par acquis
• faire qqc pour la forme
• faire qqc sans bonne volonté
• faire qqc sans conviction
• ne pas y mettre de cœur

example

• The breaking up with her boyfriend had caused in her a terrible angst, a dread of the future she couldn’t overcome; she didn’t do anything anymore in her usual way, she simply went through the motions in a sort of mental haze she couldn’t close her mind to.

• Sa rupture avec son amant avait causé en elle une peur quasi neurotique, une peur du futur quelle trouvait impossible de rejeter ; elle ne faisait plus rien de sa façon habituelle et tout dans ses actions était machinal, enveloppée qu’elle était dans une sorte de brouillard mental duquel elle ne pouvait s’extraire.

C

There is a third meaning of the exression which is quite different of the connected meanings in “A” and “B” : to simulate by gestures or movement (Oxford Dictionary). A basic translation into French of this latter would be simuler. Other possibilities are given below.

• simuler
• faire la mimique
• imiter

example

• The nurse put the dummy and utensils on the table and went through the motions of changing a baby’s diapper.

• L’infirmière mit le mannequin et les ustensiles sur la table et simula l’acte de changer la couche d’un bébé.

D

There is apparently a fourth possibility and that would be “to fake”, “to sham”, “to pretend”, which I do not get out of any source I can access but by back-translation from the reverso dictionary which gives as the French meaning faire semblant and also from the Oxford dictionary in one of its definitions, “*make a pretence**”.

• The clerk went through the motions of looking for the receipt in the file, found it but left it in the file and then told the client it was missing.

• L’employé fit semblant de chercher le reçu dans le dossier, le trouva mais le laissa dans le dossier puis dit au client qu’il manquait.

Some words that come to mind are :

• train-train – Cours ordinaire des choses, de la vie.

• routine – Acte régulier et machinal, fruit d’une habitude plus que d’une réflexion.

• nonchalance – Manière d’être ou d’agir caractérisée par l’absence d’énergie, de zèle, de soin, due à l’insouciance, l’indifférence.

• indolence – Disposition à se donner le moins de peine possible, à agir avec lenteur et mollesse.

Il m’a dit qu’il s’occupait de ma plainte, mais j’ai bien senti que c’était avec beaucoup de nonchalance.

Faire quelque chose machinalement (Larousse en ligne).

Machinalement : « De façon machinale, par habitude, sans en avoir conscience » (TLFi).

As many have pointed out, ‘just/merely/etc. going through the motions’ is negative. (So much so that ‘going through the motions’ without an adverb almost feels incomplete.) ‘Par pure inertie’ has the proper negative connotation.