It's complicated.

There are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, a relationship test and a residence test. Only a QC qualifies the taxpayer for the Earned Income Credit, the Child Tax Credit. 

Generally, you cannot just allow someone else to claim your kids. However, there are times when a child may be the "qualifying child" (QC) of more than one relative, in which case a choice may be possible. There's still an AGI test to be met (see rule 6 below).

But, in your case (you and your mom don't live in the same home) that option is not available. 

But, from what you describe, your three youngest children may actually not be your QC, but instead may be only your mom's QC. Therefor, she is the only one allowed to claim them. The most important rule is where the children lived. See rule #3. Another however is that the time at your mom's could still be considered only a temporaray absence, even if they spent more than 183 nights with her.

My advice, let her claim them if you want to, as long as they spend more than half the year with her (rule #3 is met).

A child closely related to a taxpayer can be a “Qualifying Child (QC)” dependent, regardless of the child's income, if:

1.                He is under age 19, or under 24 if a full time student for at least 5 months of the year, or  is totally & permanently disabled

2.                He did not provide more than 1/2 his own support

3.                He lived with the relative (including temporary absences) for more than half the year

4.                He is younger than the relative (not applicable for a disabled child)

5.                If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child (this essentially means that you have the parent’s permission to claim the child, if the child also lived with the parent more than half the year)

6.                If the parents of a child can claim the child as a qualifying child but no parent so claims the child, no one else can claim the child as a qualifying child unless that person's adjusted gross income (AGI) is higher than the highest AGI of any of the child's parents who can claim the child.