Let \(K\) be a fixed natural number and \({\mathcal L}\) a family of \(K\) affine subspaces of \(\mathbb{R}^d\). Let \(x\in\mathbb{R}^d\) and \(k_1,k_2,\dots\in\{1,2,\dots,k\}\) be arbitrary. Consider the sequence of projections \(z_1=P_{k_1}z,\dots,\,z_n=P_{k_n}z_{n-1}\), where \(P_k\) denotes the orthogonal projection on the \(k\)th space in \(f\). By considering the orbit of a point under any sequence of orthogonal projections on \(K\) arbitrary lines in \(\mathbb{R}^d\) and assuming that the sum of the squares of the distances of the consecutive iterates is less than \(\varepsilon\), it is shown that if \(\varepsilon\to 0\), then the diameter of the orbit tends to zero uniformly for all families \({\mathcal L}\) of a fixed number \(K\) of lines (Theorem 4.3). The authors relate this result to questions concerning convergence of products of projections on finite families of closed subspaces of \(\ell_2\) (Theorems 5.2).