Using e(+)e(-) data taken at the center-of-mass energy of 3.773 GeV with the BESIII detector, corresponding to an integrated luminosity of 20.3 fb(-1), we report the first measurement of the decay dynamics of the semileptonic decays D+(0) -> K- pi(+)pi(0(-))e(+)nu(e). The amplitude analysis gives the hadronic form factors of the semileptonic D transitions into the axial-vector meson (K) over bar (1)(1270) to be r(A) = (-11.2 +/- 1.0(stat) +/- 0.9(syst)) x 10(-2) and r(V) = (-4.3 +/- 1.0(stat) +/- 2.5(syst)) x 10(-2). This is the first in the semileptonic decays of heavy mesons into axial-vector mesons. The angular analysis yields an up-down asymmetry A(ud)' = 0.01 +/- 0.11, which is consistent with the standard model prediction. In addition, the branching fractions of D+ -> (K) over bar (1)(1270)(0)e(+)nu(e) and D-0 -> K-1(1270)(-)e+nu(e) are determined with improved precision to be (2.27 +/- 0.11(stat) +/- 0.07(syst) +/- 0.07(input)) x 10(-3) and (1.02 +/- 0.06(stat) +/- 0.06(syst) +/- 0.03(input)) x 10(-3), respectively. No significant signals of D+ -> (K) over bar (1)(1400)(0)e+nu(e) and D-0 -> (K) over bar1(1400)-e+nu(e) are observed and their branching fraction upper limits are set as 1.4 x 10(-4) and 0.7 x 10(-4) at 90% confidence level, respectively.