OK here is my take on this. In general, I do not believe children belong at weddings, funerals, or even at shul until they are able to sit quietly and without disrupting anyone. That being said, the person who brought the baby might have been very close to the niftar or to the family, and was unable to get a last-moment babysitter (funerals are SO inconvenient, aren’t they!). Weighing the pros and cons, and being dan l’kaf zechus, I think if the baby is quiet, it’s not so terrible for them, to be there, but if they start to cry or make other noises, take them out immediately. People have to use their seichel. It is NOT OK to have a baby there carrying on and upsetting the aveilim. That is inconsiderate, no matter whose baby it is. It is also very frightening for small children (though not to babies), to see adults crying and mourning. They definitely should not be taken to the cemetery, IMO.
Same goes for a wedding, IMO. I have a friend whose daughter’s chuppah was shall we say NOT enhanced by the constant squalling of the baby of a close family member who should have known better and sprung for a sitter at the wedding, AND would not take the baby out. My daughter took my grandson (my son’s child) to the back of the chuppah room after he walked down, so that if he became unruly for any reason, or bored, she could exit immediately with him. Fortunately it was not necessary, but we couldn’t predict how he would behave. People tend to forget that not everyone finds their precious darlings equally precious ALL the time, especially if they are doing what little ones predictably generally do, which is be disruptive. It is unfair to a small child to expect him to behave perfectly, and to put him in that situation is expecting too much of him. Yes, they are all cute dressed up, they are adorable when they are smiling, and it makes for nice pictures, etc. but they have their limits, and should not be placed in circumstances where they cannot be expected to behave sedately for long periods of time. If chalilah it is a funeral, not a fun experience for anyone, much less a child, keep ’em home, or bring another adult who can help out with the child, if you MUST bring him.