Why can't plasma proteins shift from capillaries to connective tissue but WBCs can be very rich in connective tissue even though obviously the WBCs had to go through capillaries. Another example: in alveolar sacs neutrophils are there in the lumen despite the presence of epithelia of alveolar sacs, and it can only reach there via capillaries. So, how can they get into lumen despite the epithelia lining? Histology textbooks say that no plasma proteins can enter or leave capillaries, but WBCs (which are much larger than proteins) can move to connective tissue via capillaries?
Cells of the endothelium are joined by tight cell junctions which are impermeable or selectively permeable. Generally, proteins can only migrate through the endothelium via active transcytosis.
Leukocytes (specifically neutrophils, lymphocytes and monocytes) express various adhesion molecules and cytokine receptors which allow them to interact with endothelial cells and facilitate their movement (diapedesis) either between (paracellular) or through (transcellular) the cells. The process of leukocytes leaving the endothelial lumen is known as extravasation.
(C,D) The process of diapedesis, whether during intravasation or extravasation, can occur by two distinct pathways: paracellular or transcellular. (C) Paracellular diapedesis. Leukocytes and endothelial cells coordinately disassemble endothelial cell-cell junctions and open up a gap between two or more endothelial cells (Muller, 2003). (D) Transcellular diapedesis. Leukocytes migrate directly through individual endothelial cells via a transient transcellular pore that leaves endothelial cell-cell junctions intact. Note that the two individual endothelial cells in C and D are distinguished by different shades of pink.
Generally, for plasma proteins to enter the tissue space, they must be able to passively diffuse out of the vessel into the tissue. This requires sufficiently large gaps between the endothelial cells of the vessel wall (and if passing into the lumen, space between the epithelial cells forming the barrier of entry into the lumen.
However, under the proper circumstances, white blood cells can actively invade the tissue across the endothelial barrier, migrate through the tissue, and may even cross over into the tissue lumen. This is a active, highly coordinated process allowing the white blood cell to bind the endothelium and push its way through a gap and allowing it to bind extracellular matrix proteins to migrate through the tissue to its destination.
Let me add that during the process of inflammation (e.g., the skin has been infected), there will be foreign materials introduced into the body which are picked up by resident macrophages and dendritic cells, which are basically waiting for this stuff to happen. Once they engulf a foreign particle, they will become activated and start releasing inflammatory molecules which attract other white blood cells to come into the said interstitial tissue (you called it CT). They come in because the capillaries in those areas become permeable to the white blood cells due to WBC-endothelial cell interactions, like adhesins and also the capillary cells (endothelial cells) may separate slightly from each other, forming little gaps, that allow proteins, more fluids to come out with the WBCs. So that's why you get swelling if you injure yourself, like if you broke your bone.. or if you got a pimple and it filled with pus (that's all the WBCs and proteins and plasma coming out of the blood vessels into the interstitium).
The reason why we want WCBs to come out during these times is because they clean up debris (macrophages), kill bacteria (neutrophils, macrophages, T-cells, etc), and the proteins may come out like antibodies, anti-bacterial proteins like complement, etc.
tldr; WBCs extravasate through capillaries in certain situations where they are needed (i.e. inflammation). The process of leukocyte (WBC) extravasation has been described by the other answers so I won't go into detail about it.