Using of Epidurals and painkillers during childbirth was not shown to have long lasting effects on the baby1.
It can affect the course of the birth, prolonging it and by slightly increasing chance for assisted or instrumental delivery, and, as some research suggest, that it may cause babies to have trouble “latching on” causing breastfeeding difficulties.
According a review by Pub Med
Any medication that a woman uses during labor enters the child’s bloodstream as well via the umbilical cord. This includes pain-relieving drugs and anesthetics delivered through epidurals. But anesthetics do not have a stronger effect on the baby than other painkillers that might be considered for use during childbirth. Epidurals have no known long-term disadvantages. One difference, though, is that births take a bit longer on average in women who have epidurals. Epidurals might make it more difficult for some babies to get into the best position for birth.
When women have an epidural, their baby is more likely to need to be delivered with the help of instruments that use vacuum suction (a “ventouse” delivery) or forceps. This is known as an assisted or instrumental delivery.
About 10 out of 100 women who do not have an epidural need an instrumental delivery, compared to
about 14 out of 100 women who have an epidural.
Before a child can be delivered using suction or forceps, an episiotomy (cut made in the back of the vagina) is usually necessary, which then needs to be stitched.
According to them it doesn't increase the chance for a c-section:
Having an epidural does not increase the likelihood of needing a Cesarean section
But other sources suggest that it can increase the likelihood of a c-section:
You might find that your epidural makes pushing more difficult and additional medications or interventions may be needed such forceps or cesarean.
source: the American Pregnancy Association
Because a standard epidural can decrease your ability to push, a forceps delivery or cesarean delivery (C-section) may sometimes be needed.
In an article by the American Pregnancy Association it's also suggested that an Epidural procedure can cause some babies to have trouble "latching on":
Though research is somewhat ambiguous, most studies suggest that some babies will have trouble “latching on” causing breastfeeding difficulties.
1 - Epidural procedure can have side effects on the mother which are
explained in the sources, but the question asks specifically about the child.