I think that Allen has adequately covered the double-heading case, where each locomotive type was crewed. As said, in the steam-plus-diesel case during the transitional era, it was often a way to continue to use existing steam power when the diesel fleet was not yet large enough to cover all assignments. I suspect that steam locomotives were often used because they were available, and not because they were optimum for the job.
Steam-plus-electric combinations were probably a lot rarer. In general there was little point in electrifying a section of railroad and then running some steam-hauled services over it, except perhaps in abnormal circumstances. But the combination did happen. I understand that there were cases in the 1960s where British Rail used it when there was a shortage of electrically heated passenger consists, and the electric locomotives involved did not have steam heating equipment. Thus double-heading with a steam locomotive was required to enable provision of steam for car heating. I suspect that some steam-plus-diesel combinations arose for the same reason.
Apparently gas turbine (GTEL)-plus-steam combinations happened on the UP. Late in the steam era, the “big” Challengers were used as pushers on the Wahsatch grade eastbound out of Ogden. This included on trains headed by GTELs, including the GTEL8500s. The big Challengers were originally designed to operated east of Green River, taking the same trains that the Big Boys brought in from Ogden. Pushing at 15 to 20 mile/h was probably not in the original design brief. I understand that pushers were used in part to allow higher upgrade speeds. Possibly they allowed higher tonnage trains, closer to what the GTEL8500s could pull east of Evanston. The Big Boy fleet was moved to Sherman Hill in the late 1940s following initial dieselization west of Laramie. There is some evidence that at least the GTEL4500s sometimes double-headed (with GTEL leading) with the Big Boys westbound up Sherman Hill.
Whilst double-heading combinations could be happenstance, MU combinations were more likely the product of specific need and engineering thought , given the complexities and costs involved.
The steam-plus-trailing diesel MU case was/is, as far as I know, confined to excursion type operations with restored/preserved steam locomotives. The steam locomotives involved were equipped to control trailing diesels via a separate controller, portable or fixed, which was not linked to the steam locomotive throttle. Apart from the difficulties of developing a suitable interface, separate diesel control suited the nature of these operations. The trailing diesel(s) were there to provide some or all of additional power on steep upgrades, dynamic braking on steep downgrades, additional power during acceleration, and for emergency power in the event of steam locomotive failure or deficiency. Otherwise they were left in idle. I understand that the first example of a steam locomotive equipped for this purpose, with a portable diesel controller was SP 4449, as part of its 1975 restoration. As a later example, in Australia, a preserved Victorian Railways R-class 4-6-4 was fitted with a diesel control stand.
The electric-plus-trailing diesel MU case was/is not all that common. That is not surprising given that once a section of railroad is electrified, it is usually advantageous to operate as much as possible of the traffic electrically, unless specific circumstances indicate otherwise.
The Milwaukee appears to have been the first to do this. In the mid-1950s, more locomotive power was required at the head end of freights in electrified territory, in a situation where obtaining additional electric locomotives was most unlikely and in fact there were factions in favour of abandonment of the electrification entirely. On the other hand diesels to assist the electrics were readily available. Thus in 1956 electrical engineer Wylie devised his eponymous throttle that allowed both synchronized and independent control of trailing diesel locomotives from the “Little Joe” electrics. This was fitted to the fleet in 1958-59. Effectively it was a miniature diesel control panel with a detachable link between its throttle handle and the electric power control handle. With the link in place, the diesel and electric power controls were synchronized. As I understand it, the diesel control panel could also control dynamic braking in trailing diesels, but whether this could be synchronized with regenerative braking on the Little Joes I don’t know. Doing that might have been a tall order, particularly given the setup procedure typically required for regenerative braking on DC locomotives. Holley recorded that additional to the Little Joes, five of the EF-5 class freight motors were also fitted with trailing diesel MU equipment, in this case done by relays rather than via a mechanical system.
Initially the diesels were used as helpers as required behind usually paired electric locomotives, but later single electrics were used more-or-less as point helpers ahead of diesel consists that originated outside of and ran through the electric sections.
The Pennsy E44 case is a bit of a mystery. Cunningham reported that the fleet had diesel MU capability. But beyond that there does not appear to be any evidence supporting that notion. Given the extent of the Pennsy’s electrified network, and its commitment at the time to electric power, it seems less likely that regular electric-plus-diesel MU operations would have been envisaged. On the other hand, diesel MU capability might have been seen as useful in unusual situations such as short-term irregular traffic peaks.
A major example, dating from 1999 and still current, is the Cape gauge Sishen-Saldanha iron ore line in South Africa. Originally diesel operated, this was electrified at 50 kV and for a while was fully operated by electric locomotives. When in 1999 it was desired to run longer trains, more head-end power was needed, but at the time the electric fleet was insufficient, so available diesels were used behind the electric locomotive consists. These were controlled by a so-called “smart cable” system that translated the electric locomotive control protocol into diesel protocol. Later Locotrol distributed power was introduced, allowing even longer trains with mid-train and rear-end locomotive consists that were electric and diesel mixes, as far as I know with the electric locomotives as Locotrol receivers. More recently the electric locomotive fleet has been renewed and enlarged, but mixed locomotive consists are still used where the total number of locomotives required for a train exceeds the maximum number of electric locomotives that the catenary and supply system can support.
In the electric-plus-diesel context should also be mentioned the British Rail Southern Region scheme of the 1960s, which allowed interworking between EMUs, electric, diesel and electro-diesel (dual-power) locomotives, with any type able to lead. Whilst it was primarily oriented to through passenger operations that traversed both electrified and non-electrified territories, there were some freight services operated with diesel-plus-electro-diesel combinations, either leading, the latter operating under electric power over the third rail, but diesel power otherwise. The diesel locomotives involved were few in number, and specially retrofitted for the purpose. It was envisaged that the scheme would also be extended to include the DEMU fleet, but this was never realized. (I think that one DEMU set was modified accordingly, but that the modifications were not actually commissioned.)
The GTEL-plus-trailing diesel case of course applied to the UP only. By the late 1950s it was looking for substantially increased power at the head-end of its freights, and the forthcoming GTEL8500 fleet addressed that need. But the existing GTEL4500s, hitherto operated singly, were not powerful enough for the emerging era. The UP did try MU operation of GTEL4500 pairs. This was problematical in that in tunnels, the trailing unit was starved of combustion air. As an alternative, it tried GTEL-diesel MU combinations, which did work well. The development work was done in 1958, and most of the GTEL4500 fleet was subsequently fitted for such operation. By way of post facto rationalization, if one assumes that through most of the 1960s, the GTELs were more economical than diesels as power for UP fast freights, then the GTEL-plus-diesel combination retained some of that economy, at the same time allowing continued use of the GTEL4500s into an era when as individual units they were insufficiently powerful, but could not be operated in pairs. And the trailing diesel locomotives were readily available from the large pool. So it was a pragmatic move. The power requirements eventually outstripped the 8500 hp of the GTEL8500s, and most of these were also equipped for MU operation with trailing diesels, in order to build combinations of up to 15 000 hp. Even at this level, the GTEL was still majority power.
Details of the GTEL-to-diesel MU interface have not surfaced. One may assume that it included appropriate mapping from the 20-notch GTEL throttle control to the 8-notch diesel control. Probably it also included provision for field loop DB control on the diesels. Mixing field-loop and potential wire DB controls was by 1958 established practice, having been originated by F-M in 1955.