The large-${q}^{2}$ behavior of the elastic form factor of a hadron or nucleus is related by dimensional counting to the number of its elementary constituents. Using the framework of a scale-invariant quark model, dimensional-scaling predictions are derived for the $\frac{B({q}^{2})}{A({q}^{2})}$ ratio in the Rosenbluth formula, multiple-photonexchange corrections, and the mass parameters which control the onset of the asymptotic power law in the meson, nucleon, and deuteron form factors. A simple "democratic chain" model predicts that for large ${q}^{2}$, $F({q}^{2})\ensuremath{\propto}{(1\ensuremath{-}\frac{{q}^{2}}{{{m}_{n}}^{2}})}^{1\ensuremath{-}n}$, where ${{m}_{n}}^{2}$ is proportional to the number of constituents $n$. In the case of nuclear targets (or systems with several scales of compositeness), we also define the "reduced" form factor ${f}_{A}({q}^{2})=\frac{{F}_{A}({q}^{2})}{\ensuremath{\Pi}{i=1}^{A}[{F}_{i}({{q}_{i}}^{2})]}$ in order to remove the minimal falloff of ${F}_{A}$ due to the nucleon form factors at ${{q}_{i}}^{2}=(\frac{{{m}_{i}}^{2}}{{{M}_{A}}^{2}}){q}^{2}$. Dimensional counting predicts ${({q}^{2})}^{A\ensuremath{-}1}{f}_{A}({q}^{2})\ensuremath{\rightarrow}\mathrm{const}$. A systematic comparison of the data for $\ensuremath{\pi}$, $p$, $n$, and deuteron form factors with the dimensional-scaling quark-model predictions is given. Predictions are made for the large-spacelike-${q}^{2}$ $^{3}\mathrm{He}$ and $\ensuremath{\alpha}$-particle form factors. We also relate the deuteron form factor to (off-shell) fixed-angle $n\ensuremath{-}p$ scattering, and show that the experimental results for ${t}^{5}{F}_{d}(t)$ are consistent with the magnitude of the $s$-wave wave function ${u}^{\ensuremath{'}}(0)$ obtained from soft-core potentials. The relation of the dynamics of an underlying six-quark state of the deuteron to the nucleon-potential and meson-exchange-current contributions is discussed. The scaling of ${q}^{2}{f}_{d}({q}^{2})$ implies that the nuclear potential (after removing the effects of nucleon structure) displays the scale-invariant behavior of a theory without a fundamental length scale. Predictions are also given for the structure functions, fragmentation, and large-angle scattering of a nucleus.