Is a woman who has difficulty finding a free moment to daven allowed to daven Shmoneh Esrei and other parts of tefilla while pushing her young child back and forth in a stroller (to help the young child fall asleep)?
While saying a bracha, and especially while davening Shemona Esrei, we are not allowed to do any actions, because it is disrespectful to be involved in anything else while talking to Hashem. Therefore, the parent should not rock the carriage then. Regarding the other parts of davening, when saying a bracha we must also abstain from doing anything else except for saying and concentrating on the bracha that we are saying. The parts of tefilla that are considered brachos are, Birkas Hashchar, Boruch Sheamar, and Yistabach, Birkas Kriyas Shema. Additionally when saying the first paragraph of Shema, (and according to some even during the second paragraph) as we are not allowed to do anything while we are declaring our allegiance to Hashem.
As a side point, if you are so busy you might be exempt from davening because you are busy caring for your children, however when you do daven, it should be done correctly.
O:CH 96-1, 191-3, M:B 96-1, 191-5. O:CH 63-6, M:B 63-18.