Mom first needs to open a USD bank account here in the DR, and it normally takes US$300 to $500 to open a DR account.
Is the USD $10k check drawn on a US account? It may be much faster to send the check for deposit in her US Chase account (using UPS or FedEx to the bank), transfer the kids $5k from there to the kids account, and then wire her money from her US account to her DR USD account.
The funds from a wire will be available to mom here in the DR much faster, but will incur a wire transfer fee and the FedEx costs. The check only need be endorsed and then written below the signature " For deposit only to Account # xxxx-xx at Chase Bank". This will protect the check in transit.
Actually, a US citizen must only declare offshore accounts if the aggregate value of all accounts outside the US is more than $10,000 at any point in a given tax year. If not reported to the Treasury (not the IRS), and prosecuted, this is a felony which can cost you your passport. (this is one of those things noted in the ICE computers at any port of entry to the US).
Here's the reporting form:
http://www.irs.gov/pub/irs-pdf/f90221.pdf
If the total value of the offshore accounts never exceed $10,000 this report is not required by US law.